Pretty - But Flawed - Finish

MY round seven game at Jersey had a pretty-looking conclusion.

Black has just played 26...Qb4xb2 in Spanton (1851) - Nigel Livesey (1753)

The game finished 27.Rh5 Kg7 28.Rxh6 Kxh6 29.Qh3+ Kg5 30.Nxh7#.
In fact, my 27th was a mistake - I should have preferred 27.Re2, and if, say, 27...Qb4, then 28.Qh3 wins.
So, the question is, how could Black have saved himself after the move in the game, 27.Rh5?
Analysis engines have no trouble finding 27...Rd6! The point is that 28.Rxh6 is met by 28...Rxf6, when 29.Rxf6 runs into 29...Qc1+, picking up the rook on h6.