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:
ולא הזכיר הכתוב טבילה באשה, כי הזכיר זוב האיש וטומאתו, ואמר בסוף (פסוק יג): ורחץ בשרו במים חיים וטהר, וחזר ואמר באשה (פסוק יט): ואשה כי תהיה זבה, כאיש הזב, דם יהיה הזוב שלה לא לובן כאיש, והזכיר הטומאה בנדה ובזבה, ואחרי כן הזכיר בזבה (פסוק כח): ואם טהרה מזובה, כאשר יטהר הזב מזובו, וספרה לה שבעת ימים, כאשר יספור הזב, ואחר תטהר, כטהרת הזב. ועל דרך הפשט, שתהיה צריכה רחיצה במים חיים כזב.
And the Torah did not mention immersion for a woman, because it already mentioned a male Zov and his impurity, and it said in the end: and he washed his flesh in living water and was purified, and then it returned and said in regards to the woman: and a woman if she has a [Niddah] issue, like a male Zov, blood will be flowing for her and not white like a man, and it mentioned the transmission of impurity in Niddah and in Zavah, and afterwards mentioned Zavah: and if she be purified from her Zovah, in the way a male Zav is purified from his Zov, and she counted seven days, just like the male Zov counts, and after is purified, like the purification of the male Zov. And the simple meaning is that she needs to immerse in living waters like a male Zov.
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.