Straight short answer to the question as phrased in title and repeated in the question body:
Hadrian had no children of his own, and thus no daughter. He also did not adopt any daughters.
From all clues considered present in the story, the emperor itself was also not even present when the executions took place, making it unlikely for his (fictional) daughter to interfere in any way. The text giving rise to the assumed reading 'must talk about Hadrian's daughter' is not a reliable historical source, but religious literature, based on some actual history, but conflating all the 'true information' — from 'time' over 'place' to 'names/identities' and more — into one narrative that deviates from the historical record.
On how to read the question
There is this text called Eleh Ezkerah which tells the story of ten men executed seemingly in very short succession under 'a Roman emperor', which from context insinuates that it must be Hadrian who the emperor/caesar refers to. Given that the text mentions 'the emperor's daughter', it is legitimate to question this daughter's very existence.
Q So which one is true?
Means in my reading of the question: one historical source with a strong religious background claims that an actual emperor's daughter was being present and interfering with an execution of a historical figure, but most other historical research excludes the very possibility of at least this part of the story to be true, as the emperor mentioned in the story had no daughter at all. So, how do we weigh the arguments?
General reliability as a historical source of Eleh Ezkerah
This religious legend is unreliable as a historical source in numerous ways. Hadrian had no daughter, Hadrian himself was not anywhere near when the Rabbi Yishmael was killed, Yishmael was killed even well before Hadrian became emperor; that is: Yishmael was killed before the bulk of the Ten Martyrs anyway. The Eleh Ezkerah is a literary source, based on some actual history, but not a straightforward historical document giving reliable historical information, but rather fictional in just too many of its details.
The biographers of Hadrian make it explicit that Hadrian remained childless
The text giving rise to this question is quite ahistorical in a few aspects. Hadrian not only 'was gay', but also 'properly married', with Sabina, but these historical possibilities seem to lead to a quickly found dead end, answering the title of this question in a short summary:
In the meantime Hadrian had experienced a personal trauma. He had been married at the age of twenty-four to a distant kinswoman, Sabina, a grand-niece of Trajan. The marriage was childless and – at least after two decades – loveless. Hadrian was in any case more interested in males.
— Anthony R. Birley: "Hadrian. The restless emperor", Routledge: London, New York, 1997.
Not every aspect of Hadrian's life can be uncovered, but in the narrative of the Eleh Ezkerah story the executions took place on one day and center around the time when Hadrian was emperor.
The clues to be read from Eleh Ezkerah
The clues from Eleh Ezkerah form irreconcilable conflicts and contradictions to established history, internally from the story itself and externally relating to other sources.
Concerning the text Midrash Eleh Ezkerah originating this story
There are conflicting accounts of his martyrdom. The Avot of Rabbi Natan states that he and Shimon ben Gamliel were decapitated in quick succession. However, the Midrash Eleh Ezkerah relates that the Caesar's daughter was so taken by Ishmael's beauty that she ordered his head be skinned while he was still alive so that she could stuff it with straw and preserve it, and this is how Ishmael died. The Eleh Ezkerah is, however, known for its use of poetic license at the expense of historical accuracy, and its author(s) probably changed the narrative in order to produce a greater effect upon the mind of the reader.
— Wikipedia: Ishmael ben Elisha ha-Kohen
The life-times of The Ten Martyrs alone should suffice to demonstrate that those people did not all die as described in the text—at the same time, nor even all when Hadrian was in charge. One of them, Simeon ben Gamliel, is for example known to have died in 70 AD, while Judah ben Dama died 136 AD, while Rabbi Akiva was dead by 135 AD. Akiva's successor Eleazar ben Shammua however outlived them all and grew quite old.
To quote Wikipedia again for the 'martydom of Yishmael':
There are conflicting accounts of his martyrdom. The Avot of Rabbi Natan states that he and Shimon ben Gamliel were decapitated in quick succession. (link) a dramatic poem (known as Eleh Ezkera) tells their story as if they were killed together. […] The Eleh Ezkerah is, however, known for its use of poetic license at the expense of historical accuracy, and its author(s) probably changed the narrative in order to produce a greater effect upon the mind of the reader.
The assumption that anything relating from that part of the Eleh Ezkerah about 'killing Yishmael and any daughter intervening' has to relate to 'Hadrian's daughter' is not very viable. According to Jewish Encyclopedia as well, the Ten Martyrs:
Among the numerous victims of the persecutions of Hadrian, tradition names ten great teachers who suffered martyrdom for having, in defiance of an edict of the Roman emperor, instructed their pupils in the Law. They are referred to in haggadic literature as the 'Asarah Haruge Malkut. Popular imagination seized upon this episode in Jewish history and embellished it with various legends relating the virtues of the martyrs and the fortitude shown by them during their execution. These legends became in the geonic period the subject of a special midrash—the Midrash 'Asarah Haruge Malkut, or Midrash Eleh Ezkerah, of which there exist four versions, each differing from the others in various points of detail (see Jellinek, "B. H." i. 64, vi, 19). Contrary to the accounts given in the Talmud and in Midrash Rab bah ('Ab. Zarah 17b, 18a; Ber. 61b; Sanh. 14a; Lam. R. ii. 2; Prov. R. i. 13), which clearly state that there were intervals between the executionsof the ten teachers, the Midrash 'Asarah Haruge Malkut, probably in order to produce a greater effect upon the mind of the reader, describes their martyrdom as occurring on the same day.
This midrash differs from the older sources in regard also to the accusation upon which they were condemned. It says that when a certain Roman emperor who had been instructed in the Law came to the Biblical passage, "And he that stealeth a man, and selleth him, or if he be found in his hand, he shall surely be put to death" (Ex. xxi. 16), he conceived the following mischievous device: He summoned Ishmael ben Elisha (perhaps the propounder of the "Thirteen Rules"; see Ab. R. N., ed. Schechter, p. 54b; comp. Ned. ix. 10), Simeon (certainly not Simeon ben Gamaliel II; see Ta'an. 29b), Ishmael, Akiba ben Joseph, Hananiah ben Teradion, Ḥuẓpit (the interpreter ["meturgeman"] of the Sanhedrin of Jamnia), Yeshebab (the secretary of the Sanhedrin), Eliezer ben Shammua' (in Midr. R. l.c. "R. Eliezer Ḥersanah," or "R. Tryphon"), Hananiah ben Ḥakinai (in Midr. R. l.c. "Judah ha-Neḥetam"),and Judah ben Baba, and demanded of them what was the punishment prescribed by the Law for stealing a man. They answered, "Death"; whereupon the emperor said, "Then prepare to die, for your ancestors [alluding to the history of Joseph and his brethren] committed such a crime, and you, as the representatives of the Jewish nation, must answer for it."
The biggest problem relating to 'Hadrian's daughter' is then of course that Hadrianic Persecutions took place without Hadrian:
But Hadrian was not going to finish the war in person. Dio reports that when not only 'all Judaea' had been stirred up but even 'the whole world', Hadrian 'then, indeed, sent against them his best generals, of whom the foremost was Julius Severus.
The story of the Ten Martyrs includes the name of the tenth martyr killed, Eleazar (ben Shammua). How the end of the Bar-Kochba-War is reported in some ancient sources, is just not reliable, with this one of the Martyrs —or one of his namesakes— meeting his end in quite a different way than in the poem, but also in relation to the emperor/Hadrian:
The decisive stage, not reported by Dio but by Eusebius, and well remembered, although in a very fanciful way, in the talmudic sources, came with a siege of the fortress of Bethar. It was 'a strong citadel', as Eusebius calls it, 6 miles (10 km) south-west of Jerusalem. The siege 'lasted a long time before the rebels were driven to desperation by famine and thirst and the instigator of their madness paid the penalty he deserved.'
A story in the rabbinic literature - which makes the siege last three-and-a-half years, perhaps by confusion with the duration of the whole war – has 'Bar Koziba' killing his uncle Rabbi Eleazar on suspicion of treachery.
'Forthwith the sins [of the people] caused Bethar to be captured. Bar Koziba was slain and his head taken to Hadrian … "Bring his body to me", he ordered.' It was found with a snake around the neck. 'Hadrian exclaimed: "If his God had not slain him, who could have overcome him?'"
No doubt it was Sextus Julius Severus, rather than Hadrian, to whom the dead leader was brought. It was probably in autumn 135, or perhaps early 136.
Again we see quite some poetic licence at work, as this might refer to "Hadrian" as something not be read literally, if one insists on doing so, but "Hadrian's representative" or something like that. But that downgrades the reliability of the information to allegory or metaphor. If one wants to apply this to the info from Eleh Ezkerah and grant that text the same loose terminology as 'still historically correct info', the slope changes from just a slippery one to a chute into a rabbit hole.
The Eleh Ezkerah as seen in its modern form & English translation
The introductory notes for a modern English translation of the Midrash based on its most widely known version and the most complete one in Adolph Jellineks Beit ha-Midrash (1853) highlight:
The idea that ten sages were collectively executed by the Roman emperor is a legend conflated from several earlier martyrological sources. Talmudic and midrashic literature both narrate the separate deaths of sages who defied the edicts promulgated by the Roman emperor Hadrian against the Jews of Palestine following the brutal defeat of the Bar Kokhba revolt in 135 C.E.; these edicts included prohibitions against the study of the Torah and the observance of Jewish law.
Some of these accounts have been incorporated into our midrash, as have different versions of the martyrdom of sages believed to have died by Roman execution. But the lives of the ten sages cited in our midrash actually span a number of generations in the first two centuries of the common era. While a list of ten martyrs and the phrase “ten martyrs” (asarah harugei malkhut) are mentioned in rabbinic texts (Lamentations Rabbah 2:2 and Midrash Psalms 9:13), the earliest narrative about their deaths to anticipate our midrash is found in the early mystical-gnostic text, Heikhalot Rabbati, which recounts the martyrdom of Rabbi Ishmael and Rabbi Akiba, two eponymous heroes of early Jewish mysticism. The scholars of today agree that the legend of the ten martyrs began to take shape within these mystical circles, and the imprint of its origins is still visible in our midrash, whose lengthiest portion is devoted to the description of Rabbi Ishmael's ascent to heaven to learn whether the decree against the sages has been issued by God.
The legend of the ten martyrs is not, however, a mystical document, not even mystical hagiography. As martyrology, it shares more in common with pagan and early Christian martyrological literature than it does with other kinds of rabbinic writings of the period. Because of its legendary character, it is impossible to say exactly when the legend took on its present form.
— Stern, see below
The translation of the relevant passages read with ample evidently visible poetic licence:
This was the order in which the sages sat in pairs: Rabbi Ishmael and Rabbi Simeon Ben Gamaliel; Rabbi Akiba and Rabbi Hanina Ben Teradion; Rabbi Eleazer Ben Shammlja and Rabbi Yeshivav The Scribe; Rabbi Hanina Ben Hakhinai and Rabbi Yuda Ben Baba; Rabbi Hutzapit The Translator and Rabbi Yuda Ben Dema.
The Roman emperor now entered, followed by all the noblemen of Rome. He addressed the sages, “Who shall be executed first?” […]
So Rabbi Ishmael mourned and wept over the body of Rabbi Simeon
ben Gamaliel. The emperor said to him, “What is this? You, an old man, weeping over your fellow sage! You should be weeping over yourself!”
Rabbi Ishmael replied, “I am weeping over myself! For my fellow sage was greater than I in knowledge of Torah and wisdom. And now he has joined the heavenly academy before me. This is why I am weeping.”
While Rabbi Ishmael was still giving voice to his grief, weeping as he lamented, the daughter of the emperor looked out her window and saw how handsome Rabbi Ishmael the high priest was. Compassion for him seized her, and she sent to her father asking him to grant her one request. He sent back to her, “My daughter, whatever you ask I shall do—so long as it does not concern Rabbi Ishmael and his colleagues.” She sent back, “I request you to allow him to live.” The emperor replied, “I have already taken an oath.” She responded, “Let me then request that you order the skin stripped off his face, that I may have it to use instead of a mirror to look upon myself.
— David Stern: "7. Midrash Eleh Ezkerah or The Legend of the Ten Martyrs", in: Victoria Clark, David Stern & Mark Jay Mirsky (eds): "Rabbinic Fantasies. Imaginative Narratives from Classical Hebrew Literature", Varda Books: Skokie, 2001. doi (A version of the 'Hekhalot Rabbati' to compare is here)