Regarding your initial assumptions:
- I recall a kina in which the children of R' Yishmael (haKohen Gadol) are put in a dark room together and forced to have relations.
As noted by @kouty there is no indication that they actually had relations.
- At the end of the kina they recognize each other and kill themselves.
As noted by @kouty there is no indication that they kill themselves.
That being said, the question still stands:
If a person is taken and forced to have illicit relations (without the option of death) should they actually take their own life after the fact?
Regarding forced relations in particular, this likely does not even enter the discussion of yehareg v'al yaavor, since it the victim is a passive participant, then the victim is not considered to have performed any act; sin, or not. (Cf. Sanhedrin 74a).
That being said, the question still stands: is suicide permissible, or even encouraged after performing a cardinal sin?
From everything we know from the rulings of the Talmud and Geonim, suicide is always forbidden (cf. Masechet Semachot ch. 2, Maimonides' Hilchot Avel 1:11, and Hilchot Avel 5:1-4).
Probably influenced by certain tragic events in the First Crusade, certain Tosafists (Sefer Mitzvot Katan #3, see here) decided on the basis of questionable aggadic evidence that preemptive suicide is actually permitted to prevent one from violating cardinal sins.
None of this pertains to suicide after a sin.
Although one can find a source for anything "in Judaism", i.e. a great Jew who may have held an opinion, that does not mean that that is reflective of the Jewish position on the whole.
The lone view in this case is the Yaavetz who could be interpreted as permittng suicide after performing both cardinal sins, and any sins for which one is liable to the death penalty. (It should be noted that this came up as part of his primary discussion of abortion, which is much more lenient than general murder. His emphasis on the various views on abortion indicates that he likely does not really permit suicide in normal cases. Furthermore, it is not clear to me, how much of what he is saying is a limmud zechut; post hoc justification, and how much of it is l'chatchila)
However, whether he intended this or not, this wild view was rejected in the strongest possible terms by countless successors.
For example, R. Moshe Feinstein writes in Igrot Moshe (CHM 2:69):
והנה בשאילת יעב"ץ ח"א סימן מ"ג ראיתי דברים שלא נתנו להאמר...וממש דברים בטלים הם אף שכתב זה אדם גדול כהריעב"ץ,
He describes the statement of Yaavetz, as one that is forbidden to even be mentioned, which constitutes utter "devarim b'telim" a derisive term for useless words.
Accordingly, in no uncertain terms, regardless of the dispute regarding preemptive suicide, it is unquestionably a grievous sin to commit suicide after one has committed a cardinal sin (involving incest, or anything else), a sin liable to death, or any other sin.
It would be an ironic double tragedy, for as noted by R. Yechiel Michel Tukatchinsky (Gesher HaChayim ch. 25) death usually brings atonement for sins. How tragic would it be if the death itself was self-inflicted, thus failing to bring atonement and indeed incurring further punishment.
As an aside obviously many if not most suicides are the result of mental illness. In such circumstances, the one who commits suicide suffers from mental illness, and inasmuch as he was not in control of himself, he would not be considered the perpetrator, but the victim.
