As Japan does not seem to have been especially wealthy at the time and would thus not have provided anything like the economic benefits to the Mongols that southern China could have, what convinced Kublai Khan to invade Japan first? Was the Korean King Chungnyeol instrumental in this decision?
There are 2 main parts to this question. The easier one is: why did Kublai Khan invade Japan before first conquering the Song Dynasty (southern China)? The answer to this 1st question gets a bit lengthy (but not too much, I hope)
As regards the 2nd question, the role of Prince Sim (諶), (posthumous title King Chungnyeol, 1274–1308) in the invasion of Japan. He was the son of King Wonjong.
1. Why did Kublai Khan invade Japan in 1274 (Bun'ei) & 1281 (Kōan)
The context of this question is, since Kublai had not completed subjugating the Southern Song Dynasty, which he did in the final battles of 1276-9, why the rush to invade Japan? What was the impetus?
The straight-forward answer is that it was to strangle the economy of the Song Dynasty because Japan was a valuable trade partner of the Southern Song Dynasty. Also, the maritime trade route with China did not have to go through Korea/Goryeo (which had already submitted to the Yuan Dynasty).
The failure to establish a positive relationship with the Japanese resulted in the invasions of 1274 & 1281.
Song coins and trade with China
In fact, China's maritime trade with Japan was so strong that the (Japanese) Kamakura bakufu suffered somewhat with debt and inflation (especially from importation of Song coins). From The Cambridge History of Japan, Volume 3 - Medieval Japan (1990), pp. 358-360:
Throughout the late Heian and early Kamakura periods, trade with the continent - Sung and Southern Sung - continued, and two results of this trade are especially significant ... One was that trade with China offered elites many products unobtainable in Japan; the other was that the trade brought in a large quantity of Sung coins. The latter was by far more important because it led to the monetization of the Japanese economy, which in turn had profound effects on the political, economic, and social history of Japan during the Kamakura and Muromachi periods. ...
Foreseeing the various difficulties to arise as a result of the economy's continuing monetization, in 1239 the bakufu was forced to issue a decree prohibiting the appointment of monks, merchants, and moneylenders to the position of deputy jito (jito-dai). Some in the warrior class had begun to appoint these men to collect dues directly from the shoen so that those who had lent money against the warriors' land could be repaid. To prevent such a development, the bakufu in 1240 admonished its retainers living in Kyoto not to indulge in luxury beyond their means. These developments, however, were only the harbinger of the much more serious consequences of monetization.
Importance of Japan to conquering the Song Dynasty
Kublai's invasion of Japan was not knee-jerk, nor taken lightly. As mentioned, it was a strategic decision taken only after numerous attempts at establishing a subordinate (tributary) relationship with Japan had failed. In my mind, King Chungyul (or Chungnyeol) was not involved in the decision-making (more below).
In total, Kublai sent emissaries to Japan six times, beginning with the first letter in 1266 (arrived 1268). A lot turns on the intention of Kublai and, as Gavin has shown in his answer, it depends on how the initial (1266) demand letter is interpreted. Was it a threat or even declaration of war? In fact, it could be a routine request or, at least, non-threatening. Kublai had friendly intentions with Japan at the beginning because the prize was always the Song Dynasty.
Again, from volume 3, pp. 414-415 (emphasis mine):
A document from the Mongols (dated 8/1266 and arriving in Japan on 1/1268) first described the Mongols' might and then recounted how they had brought peace to Koryo, explaining that the Mongol-Koryo relationship was like lord and vassal or father and son: Koryo was the Mongols' "eastern vassal state." Japan is close to Koryo, the document continued and, since Japan's founding, had normally had exchanges with China but had not yet concluded peace with it. Saying that perhaps "your honorable country" is unaware of these circumstances, the document expressed the Mongols' wish to establish friendship between the two states. The wording was courteous and did not demand Japan's submission, but it concluded that if there were no other way, the Mongols would resort to force of arms. Reminiscent of the edicts issued by the emperor in the Han Chinese state, this document coupled the Mongol sense of superiority with a Chinese moral view, following traditional Chinese foreign-policy forms. From ancient times China had, in return for the tribute goods that neighboring kingdoms brought and presented at the Chinese court, granted these states "gifts" of a higher value than the tribute, preserving a China- centered foreign-policy structure. Ever mindful of Chinese tradition, Kublai naturally patterned his actions after that policy. The type of relationship that Kublai first sought with Japan was probably this sort of tribute-gift exchange, one that combined politics, foreign relations, and economics.
Why did Kublai seek peace with Japan in 1266? The Mongols were preparing to attack the Southern Sung, and Kublai, judging the Sung to be strong, proceeded carefully with his attack plans. Part of these plans required strengthening control over Koryo. In his document Kublai referred to Koryo as the Mongols' "eastern vassal state." With Kublai's military backing, Koryo's King Wonjong had suppressed the military officials who had previously controlled national politics and had increased his own authority as sovereign. Thus, from Kublai's point of view, Koryo was indeed a newly attached country. Using the negotiations with Japan as a tool, Kublai was able to intensify his domination over Koryo. Also, in preparing to invade Southern Sung, Kublai had come to realize that Japan, even though it was a small eastern island really not worth taking, it could not be left alone. Japan traded with the Southern Sung and therefore continuously added to the Sung's financial strength. Seen in this light, Japan had to be cut off from the Sung and added to Kublai's empire. And this would be possible if Koryo were used as a go-between.
Finally, the Mongol tradition of valuing their emissaries should not be underestimated. I cannot remember how many, but some of the emissaries to Japan died. This was clearly against their strict tradition. Just as the last Shah of the Khwarezmian dynasty found out, disrespecting by shaving the heads of Genghiz Khan's two ambassadors and killing one (he was the only Muslim) led to Genghiz Khan's conquest of Khwarezmia. Japan's warrior class probably did not appreciate this aspect of Mongol diplomacy, which led to the two invasions.
2. Goryeo's Prince Sim (after 1270, Crown Prince) & Mongol invasion of Japan
Simply put, Goryeo was not significant at all to the Yuan Dynasty except as a military base to the Mongols, i.e. launching pad for the invasion of Japan. Obviously, I have no evidence that Prince Sim/Chungyul did not influence Kublai Khan to invade Japan. But, I can offer three reasons why Chungyul was "not instrumental" to the invasion of Japan:
Kublai Khan was an excellent strategist. Not sure he needed help
from Prince Sim (and a non-Chinggisid at that). Kublai's civil war with his youngest brother, Arigh Borke, is clear testament to his ability as he fought a near perfect campaign (a lot of troop maneuvering in China's Northern Plain and Mongolia). For reference, see operational level of war & a report by the U.S. Army's School of Advanced Military Studies (SAMS) on Mongol Warfare - PDF).
Prince Sim was sent to Kublai's court whilst his father, Wonjong, was still the Crown Prince being held hostage back in Goryeo (by the Choe military regime). In 1270, King Wonjong finally ascended the throne, allowing Prince Sim to be made Crown Prince. That's why the marriage should be 1271 onwards (re Gavin's answer). In any case, Crown Prince Sim was of no significant (neither diplomatic nor military) value during his time at the Yuan court because even his father, King Wonjong, needed the help of the Mongols to (in effect) ascend the throne.
Goryeo was one of Yuan Dynasty's many administrative centers, not that valuable other than a launchpad for the invasion. See Zhengdong Branch Secretariat, only useful as a vassal (tribute) and as "Japanese Expedition province".