I believe "日本" and its derivatives (like "日本人" and "日本語") are actually the only words where "日" is read as "ni." On the Japanese version of Wiktionary it's explained like this: "呉音読みの「にっぽん」から促音を略した「にほん」とも発音されるようになった。"
In simple words, it was pronounced as "Nippon" (which is the regular reading of "日") at first but then was simplified to "Nihon." Thus, you can consider this reading an exception you won't encounter anywhere else.
The kanji 日 (nichi) is still only pronounced as (nichi).
However, when combined with other kanji, the sound becomes truncated.
In the case of 日本, Shuuichi73 has explained above. The sound evolved from /nip・pon/ to /ni・hon/.
日本 (nihon)： /nichi・hon/ → /nip・pon/ → /ni・hon/
You can see other examples of how this works:
日記 (nikki)： /nichi・ki/ → /nik・ki/
日光 (nikkou)： /nichi・kou/ → /nik・kou/
日程 (nittei)： /nichi・tei/ → /nit・tei/
To Christian&Shuuichi73 Thank you both so much for using your time which you'll never get back for explaining this to me! ありがとうございます！！！！w^ :3
Ni?
In some words like Nihongo, Nichi is readen as Ni eventhough it says that there's no reading of Nichi as Ni! Then why is it readen as Ni?! (Sorry fot my weak English)