What should I have done, Kh2 at ...36?
Honestly I think white has too many draw threats.
36... Kh7 37. Qf4! Bd4 38. Qf5+...
I think your best chance at a win was 36... Qd4!
This threatens 37... Rxf1 38. Kxf1 Qd1#
It also wins whites powerful d pawn. The best response for white in this situation is the following line.
36... Qd4 37. Qc8+ Kh7 38. Qc2+ g6 39. Kh1
Here is a diagram of the position.
Here black has some mating threats with Be5 followed by Qh4. Or you can play for the end game by simply picking off whites pawns.
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