I drank a whisky soda last night while I was watching soccer game on TV.

I drank a whisky soda last night while I was watching a soccer game on TV.

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However, I fell asleep while I was watching it.

"felt" is the past tense of "feel". "fell" is the past tense of "fall", and the phrase is "to fall asleep".

That’s why I didn’t know that which one won.

It would be better to say "which team won". "One" is an indefinite pronoun so unless you said "team" earlier in the sentence, you should not use it.

I want to Japanese national team was won. Either: I wish the Japanese national team had won. Or: I wanted the Japanese national team to win. Or, if you don't know which team won: I hope the Japanese national team won.

"was won" is passive, you want active here.

They are the national team of for the London Olympics.

So I felt a little bit good better this morning.

better: more good. Or you could say I felt "okay" this morning, which means between bad and good.

I wanted to more sleep but I commuted to the office.

Either: "I wanted to sleep more" or "I wanted more sleep". The word "sleep" is both a noun and a verb.