My intention is to accept this linked answer, but to have supporting evidence and further information provided in this one based on research and investigations done after I was pointed in the right direction.
My narrative is based on 'Defeat in ABDA' except where otherwise noted.
Doorman and Helfrich met, or at least talked, several times in the lead-up to the Battle of the Java Sea. One of the earlier cruises where Doorman had sighted the enemy had ended in failure due to Helfrich's suggestion of 'attacking in waves', and it is possible that Doorman wanted to attack 'properly' this time round and to get at the convoy. It is noteworthy that Helfrich was upset at the ships which came out of Java Sea without being sunk, it having been his expectation that any engagement would be to the death (even for no greater purpose).
Meanwhile, questions about Allied integrity should be put to rest. While Churchill and Wavell conspired to redirect troops from Java to Sri Lanka and Burma in mid-February, the opposition of Curtin to this stopped such proceedings. Further, British ships were left under Dutch command when the Allies decided to fight for Java as long as possible.
Perhaps another thing to consider is that Helfrich's reputation in post-war Indonesia was quite poor as evidenced by the British officer, Laurens van der Post, who considered the Admiral:
"a presumptuous boaster and a natural bully," whose actual battle experiences consisted of nothing but verbal ones.
Around the 13th and 14th February, Doorman and Helfrich spoke and reached a decision:
...to go northwards through Gaspar Strait, round Banka, and back through Banka Strait, "destroying any enemy forces seen".
This excursion failed due to Allied air support being non-existent, and Doorman returned to Batavia. Japanese shadowed the navy and kept it under constant air attacks for six hours. When Doorman returned, he had to split his fleet because there were too many ships in port:
Because of the congestion at Tanjong Priok (where on 16th February every berth was crowded and at least thirty ships were anchored in the roads) the striking force was split up between Java and Sumatra on its return. "Hobart, Exeter, Tromp, Barker and Bulmer followed De Ruyter in to Tanjong Priok and anchored at 9.52 a.m. on the 16th," recorded Hobart. That morning too, the rest of the American destroyers arrived at Ratai Bay to fuel.
Nevertheless, that sally by Doorman had delayed the arrival of the Japanese convoy by a day. The decision to withdraw was justified as due to Japanese air superiority that would have determined any naval engagements. The next move Doorman undertook was to head east to protect Bali and Timor. On the 17th Feb, Doorman and Helfrich met again:
Admiral Helfrich and Rear-Admiral Doorman talked together at Helfrich's headquarters. It was decided to concentrate in the east at once to contest the assault on Bali. De Ruyter, Java, Piet Hein, Kortenaer, and the American destroyers Ford and Pope, would go via the Indian Ocean and Tjilatjap; and Tromp, with the four American destroyers from Ratai Bay—Stewart, Parrott, Edwards and Pillsbury—via the Java Sea and Surabaya. Helfrich, knowing that time was the vital factor and that the hours consumed in taking the Striking Force from western to eastern Java would possibly make its arrival at the scene of action again too late, told Doorman not to wait until he had concentrated completely, but to make a series of raids in successive "waves" (if that were necessary) through Lombok Strait. In addition, a suggestion by the Dutch naval commander at Surabaya was accepted, that motor torpedo boats from that port should follow up the attacks by the main force, and exploit the enemy confusion.
Doorman sailed at 10pm on the 18th, and moved along the south coast of Java to Bali. Kortenaer had been left behind as it ran aground. His force made it to Bali by the evening of the 19th, engaged in combat in which Piet Hein was lost, and then disengaged. Dutch cruisers headed to Surabaya; American destroyers to Tjilatjap. Another 'wave' engaged for a hit-and-run attack but disengaged similarly quickly.
Doorman tasted increasingly the bitter fruits of lack of tactical exercises and of adequate communication facilities in his heterogeneous force, and of misleading
intelligence. It would seem that, based on the original reconnaissance reports, and on those of the bomber pilots who attacked the Japanese ships during the 19th, it was believed that numerically superior and more powerful Japanese forces would be met. This belief persisted after the action, and there would seem to be little doubt conditioned Doorman's mind and those of his commanding officers to an extent which gave the "tip-and-run" aspect of the attack undue prominence.
The Allied waves had outnumbered the enemy, but failed to make their numbers count. Tromp, in the rear of the second wave, had to be sent to Australia for repairs. The Japanese conquered Timor, launched a bombing raid against Darwin, and were in the final stage of preparing to attack Java.
On the 20th February, the day after the first Darwin air raids, the Combined Chiefs of Staff in Washington told General Wavell that Java should be defended with the utmost resolution by all available combatant troops. "Every day gained is of importance. There should be no withdrawal of troops or air forces of any nationality, and no surrender." He was told to augment the defence of Java with available naval forces, and with U.S. aircraft at his disposal assembling in Australia, but that Burma had been removed from his command and once again placed under the control of India, and that land reinforcements on the way from the west would not go to Java but were being diverted to augment the defence of Burma, Ceylon, and Australia, points vital to the continuance of the war against Japan.
Admiral Helfrich stressed to General Wavell the impossibility of defending Java with his existing (and diminishing) naval forces. He asked that, if reinforcements could not be sent, the Anzac Force from the east and the Eastern Fleet from the west should carry out diversionary raids or demonstrations into or towards the Java Sea and China Sea to ease the situation. General Wavell was unable to grant this request because of the enemy's massive air superiority, which it was beyond Allied power to counter.
Now, because Helfrich was unsure where on Java the Japanese would invade, he divided his naval forces into Eastern and Western Striking Groups. He also issued a specific message to all ships under his command:
I inform all officers and ships' companies that the situation is critical. I wish to
impress upon all of you the necessity for every effort against the enemy to prevent
his landing on Java. Every opportunity for offensive action must be seized and all
sacrifices must be made to this end.
That same day, General Wavell cabled Churchill:
I am afraid that the defence of ABDA Area has broken down and the defence
of Java cannot now last long. It always hinged on the air battle. ... Anything put
into Java now can do little to prolong struggle. ... I see little further usefulness
for this H.Q.
Churchill had on the previous day (20th Feb) already taken a decision to divert some Australian troops heading to Java to Burma instead. Wavell and Churchill had previously discussed the futility of defending Java, and this move should be taken in that context. Roosevelt also supported diverting the Australians into Burma, but Curtin refused this high-handed move. On the 23rd, Churchill again set the Australian convoy to proceed homewards (by no longer to Java).
On the 25nd Feb, the ABDA command was dissolved with the intention of preserving morale:
On 22nd February Wavell discussed the withdrawal of his headquarters from Java with the Governor-General of the Netherlands East Indies, Jonkheer Dr van Starkenborgh Stachouwer, who thought that withdrawal after invasion would damage public morale, and that, as an alternative command organisation was already in existence, it would be better that the ABDA Command should be dissolved, and not withdrawn. Wavell agreed, and in a message to the Chiefs of Staff advanced this view, pointing out that since the control of Burma had reverted to India, ABDA held little to command, and that the local defence of Java could be better exercised under the original Dutch organisation. ... In Java, the British forces now came under the orders of the Dutch Commanders-in-Chief: Vice-Admiral Helfrich, who ceased to be "Abdafloat", and resumed his former Dutch title of Commandant der Zeemacht ("C.Z.M.")...
The same day, a definite threat materialized in the Macassar Strait and Helfrich ordered all cruisers and destroyers to reinforce Doorman at Surabaya. At this stage, all Allied commanders seem to have been on the same page about a 'fight to the death' for Java alongside the Dutch. The British hoped to use that time in order to strengthen defenses of Burma, India, and Sri Lanka.
Doorman engaged in cruising during the 25th and 26th. On the 27th, Doorman messaged Helfrich at about noon:
This day the personnel reached the limit of endurance; tomorrow, the limit will be exceeded.
The Western Striking Force had meanwhile been at sea as well. Helfrich didn't expect a battle near Surabaya, and didn't order the Western Striking Force to join Doorman in the early part of the 27th (leading to their exclusion from the Battle of the Java Sea). Meanwhile, that same morning the USS Langley, an aircraft tender, had been destroyed by Japanese aircraft on an approach to Java from the south after having detached from convoy MS.5.
As Doorman entered Surabaya after 2pm on the 27th, he received a note from Helfrich which related sightings of the Japanese to the south-east. Doorman turned and proceeded to sea. The Japanese were observing Allied movements, and three destroyer flotillas gathered while the Japanese convoy reversed course.
These engagements led to the Battle of the Java Sea. That entire battle, Doorman tried to search for the Japanese convoy and to cause damage to it. This was a blind search while the Allied fleet was fully scouted by Japanese aircraft. De Ruyter, Doorman's flagship, was sunk at about 11pm. The standing verbal instructions were that "any ship disabled must be "left to the mercy of the enemy"."
Therefore, Waller, in command of Houston and the remaining senior officer, turned to disengage and to steam to Batavia. Waller reported the destruction of De Ruyter and Java on 00:50 28th Feb. He justified the disengagement by the loss of two cruisers while enemy forces were near-intact and had air superiority. Helfrich later criticized this move in the Dutch Official Report of the Battle of the Java Sea:
Strictly speaking the return of Perth and Houston was against my order 2055/26—"You must continue attacks till enemy is destroyed." This signal was intended to make it quite clear that I wanted the Combined Striking Force to continue action whatever the cost, and till the bitter end. Perth did receive this signal. Both cruisers were undamaged [Houston's after triple turret was out of action] and it was not right to say in anticipation "It is no use to continue action", considering the damage inflicted upon the enemy cruisers, which in my opinion must have been severe . [Actually the enemy cruisers were all in battle trim.] However, it is possible that other facts had to be considered, such as shortage of fuel or ammunition. [Houston, as stated above, had very little ammunition remaining.] The decision of the captain of Perth is even more regrettable as, after all, both cruisers did meet their end. Probably on the night of 27th-28th February they would have sold their lives at greater cost to the enemy.
The overall delay of the invasion of Java was one day while the cost to the Allies were nearly all of their surface fleet, including their experienced crews.