Unlike other ritual impurities, which explicitly list various immersion or washing requirements, biblical niddah has no such explicit requirement in the written Torah ( to the best of my knowledge ) so from where do we learn that a biblical niddah is required to immerse on the eighth night after getting her period ( assuming of course that she's no longer bleeding )?
There truly is no clear source for this. There are several different opinions in Rishonim, each one has its limitations.
The simplest source is a Kal Vechomer, whoever touches what she lies or sits on has to immerse to be pure, how much more so herself. The Rambam (Issurei Biah 4:3), quoting the Sifra [according to the Maggid Mishna, in a quick look I couldn't find it saying that about the Posuk, only that it means that she and he have the same Mikvah requirements], says it is a Binyan Av. Tosfos gives some other opinions (see the Hagaos Maymonis on the Rambam there).
Also I would note that there is no such thing as a ritual impurity that goes away due to the passage of time alone. Some require Mikvah, some require other things instead of or in addition to the Mikvah. But there isn't one that just "expires." There is no reason to think that this is an exception.
Regarding Talmudic level sources, you can look at Avodah Zarah 75b where it says that מי נדה means that the water that Keilim are immersed in has to be the same shiur as the water used for a Niddah - 40 Soah. Also Shabbos 64b where it learns that והדוה בנדתה means that she has to go to the Mikvah to be permitted to her husband (see Rashi there). These sources are brought in Tosfos on Yevamos 47b (as well as the Kal Vechomer above) where it discusses explicitly that a Ger and a freed Eved immerse in the same type of Mikvah as a Niddah and have the same requirement regarding any separation from the Mikvah water, where Rashi discusses that the similarity for all three is that the purpose of the Tevilah is not for the ability to handle Taharos.
I found another source in Ramban's commentary to the Torah ( ויקרא פרק טו ) for why this is done:
Basically, the female illness of a Zavah is parallel to the male illness of a Zov ( he says this elsewhere ), and since it already mentions immersion for the man it should be clear that it's needed for the woman also. And since the laws regarding transmission of ritual impurity are the same for a Zavah as for a Niddah, then if the Zavah requires immersion then Niddah must also require immersion. It's a transitive relationship in regards to immersion between the three, Zov, Zavah and Niddah.