In your starting position (which Black wins with the move), it only takes Stockfish 13 a few seconds to find that 43. Kd3 draws for White. In the line you give, 44...h4 is Black's only move to avoid losing, but then 45. e5+ is White's only move to draw, which Stockfish sees at a depth of 20 after 7 seconds.
It's hard to believe that Black was supposed to win this position with White to move. I couldn't locate the puzzle on this site. Is there a way to find a puzzle by its number?
edit: formatting, adding a link to the puzzle