There is no feeling in the world that can compare to how badly I felt that day.

No, it does not work (at least the way I think you want it to work).
Badly is an adverb. Adverbs modify verbs.
Bad is an adjective. Adjectives modify nouns.

I felt badly. - This means you were not very good at emotions or feeling things. You were bad at feeling.
I felt bad. - This means your emotions were involved and you felt sad or guilty.

Also, take heart that even native speakers get this wrong.

Donald Trump once corrected Cindi Lauper. She said, "I felt bad." He said, "No, you felt badly." She said, "Fine, I feel badly, whatever." He was wrong to correct her.

Thanks!

So, is this 100% fine?

There is no feeling in the world that can compare to how bad I felt that day.

Tom

PS: How often do we use "I felt/feel badly."?

Mr. TomThanks!

So, is this 100% fine?

There is no feeling in the world that can compare to how bad I felt that day.

Tom

PS: How often do we use "I felt/feel badly."?

Yeah, it's completely fine to say. The writing isn't great, but grammatically it's okay. The only bad thing about it is that it's vague and doesn't really des

(And no one uses "I feel/felt badly" because no one thinks they are bad at feeling, unless they have mental or physical problems that prevent them from actually feeling anything.)