"Mir ist besser."

Translation: I feel better.

[US2Germany]

Does "Ich bin besser" work here too?

[Menschenkind]

No, that means 'I am better', comparing yourself to someone else. I find 'Ich fühle mich besser' or 'Es geht mir besser' more appropriate here. 'Mir ist besser' sounds rather uncommon.

[xInfx]

I'm German and I never really heard someone say 'Mir ist besser'. As you said, 'Ich fühle mich besser' or 'Es geht mir besser' would be more appropriate here.