I captured the bishop on e6 for the second move. The bishop was hanging. The mistake was that my bishop on g5 was also hanging, and the exchange would have left White with a pawn material advantage and a favorable position for eventual pawn promotion. Moving the king to h8 still leaves the bishop on g5 protected and attacking White's queen on h6, forcing White to move the queen to h3 if he wants to keep his queen.
Hanging pieces puzzle
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I'll explain why I missed this one in a reply below in a bit because I don't want to spoil the puzzle for anybody.