I would like to know whether the sentence using "which" would be correct.
I'm so tired, the reason for which I don't know.
The construction is just like
''The building was destroyed in a fire, the cause of which was never established.''
I know this using:
I don't know the reason why I'm so tired.
Your sentence I'm so tired, the reason for which I don't know." is correct and grammatical.
However, the phrase "the reason for which" is usually only used in formal speech, not in every-day conversation. Most of the uses of "which" that you are after are regarded as more formal ways of phrasing things. If you are trying to use formal language on purpose, keep in mind that contractions like "don't" and "can't" are also themselves informal. If you want to go all the way with formality, I would recommend: "I am quite tired, the reason for which I do not know."
More normal speech would be "I'm so tired and I don't know why.". I don't think you were asking about that, but it is a good way of phrasing things. Adding "...and I don't know why." is a simple and versatile way to express uncertainty without having to rearrange the structure of a sentence. It is not formal though.
