The first point to note is that after the natural 10.Kxh2, Black has perpetual check with the queen at h4 and f2. This gives White a choice of 10. Kf1 or 10. Kh1.
After 10. Kh1, Black must threaten checkmate with 10...Qh4 (or her pieces all start falling off the board). Then the only way out for White is 11. Rxe4+, to remove Black's control over f2. (Discovered checks will only allow Black to unpin her knight with fatal consequences.) Black must recapture somehow - admittedly, recaptures cannot be taken for granted in this variation, but this time Blacks queen is attacked. There's no point getting greedy with Qxe4 attacking a4, because White can drive the queen off with Nc3, then play Ne5+ and Nf3 to smoke out the bishop. Instead, 11...dxe4 covers f3. The dénouement (all forced) is: 12. Qd8+ Qxd8 13. Nxd8 Kxd8 14. Kxh2 - and nominal material balance has been restored, although the endgame is thought to favour White. Note in this position, white threatens Bb3, targeting black's f7 pawn. A naive attempt to handle it with 14...f5?? 15. Bg5# is to be avoided. Instead, black should play 14...Be6.
After 10. Kf1, 10...Qh4 again threatens mate. The simple Be3 allows ...O-O! Suddenly all Whites threats are neutralised and Black has two clear pawns for the piece with huge pressure on Whites kingside. Better is 11. Nd4+ b5 and only then 12. Be3. The knight threatens to come back to f3 kicking the queen and exchanging off Blacks useful bishop, and combined with the fact that Qxd5 forks the e4-knight and a8-rook this makes 12...bxa4 a bad idea. 12...Bg4 is much more promising - Black looks to have good chances after 13. Nf3 Bxf3 14. gxf3 Qh3+ 15. Ke2 O-O-O! - which suggests that White should go with the risk-free edge after 10. Kh1.