Hello! I can certainly try, but I’m sorry to say that as with tons of other Japanese words, there simply isn’t a hard-and-fast rule I can teach you.

「ひ/び」 is used for days of the week and words like 誕生日｛たんじょうび｝. It is also used for holidays, like for instance　検漏感謝の日｛けんろうかんしゃのひ｝or 成人の日｛せいじんのひ｝.

「か」 is, as far as I know, only used for certain dates of the month. The first 10 days of the month are counted using the Japanese pronunciation system （ひとつ、ふたつ…）, with the exception of １日, which is pronounced ｛ついたち｝. I know, fun! So, for the dates of the month, it goes ２日｛ふつか｝、３日｛みっか｝, and so on. This goes on up to １０日｛とおか｝, and then it continues with a few special dates. １４日 is ｛じゅうよっか｝, ２４日｛にじゅうよっか｝, and then ２０日, which is pronounced ｛はつか｝, because it wasn’t crazy enough! Good luck with that.

「にち」 is, as far as I know, just ‘day’. It’s the on’yomi of 日. This pronunciation is part of compound words like 一日中｛いちにちじゅう｝.

Cats of Furin in “chic ribbon outfits” (as the description on the website says) 🐱🎀

Little Sakura wants to be friends with the fish 🐠