After Black plays 20. ...Kg6, I can't tell if I should just take the rook and threaten the pawn on g7, or if I should attempt checkmate right away by playing Qg7+. I can't see a line that guarantees me a checkmate from here, but at the same time, Black's rook does not look all that tempting to take because my Queen abandons many of White's vulnerable pieces by being isolated in the corner. I can't think of a good variation that works without Black blundering anymore than they already have. Help, anyone?
I would play Qg3 without even thinking. Black has to reply Qg4 and you trade the Queens and win. Qxh8 also wins, but you have to calculate the consequences of a series of checks and it's possible to miss something. When you have a simple win there's no reason to complicate it (in reality Qxh8 is also quite simple to calculate, but it's a good general principle not to go for unnecessary complications in a winning position).
Not sure what the problem is but it should be fairy easy to see that white can win with about any move. Taking the rook seems simplest to me, even if Houdini thinks 1.Qg3+ Kf5 2.Qxgy or 1-Bg4 2.f3 is stronger.