The first thing to notice is that 1.Nd3 is not mate because of Kxf3, but it would be mate if the f-pawn were defended. The second thing is that 1.Nd5+ Kf5 2.Ne7+ Kf6 3.Rg6 is mate, as is 2...Kf4 3.Ng6+ Kf5 4.Nh4+ Kf4 5.Nd3 mate. The only problem there comes after 2...Ke6, when Black escapes. But if the e6 square were blocked, it would all work well.
Finally, it is clear that White must do something quickly, or the black h-pawn will queen. Since 1.Rxh2 loses White's bind and lets Black escape with 1...Kg5, the first move must create a mating threat.
Combining those ideas, you should come up with 1.Ke2, defending f3 and threatening 2.Nd3 mate, but what to do after 1...Ba6+? After 2.Kf2, the black pawn promotes to a knight with check. Well the pawns on b4 and c3 must be there for a reason, so let's try 2.b5 Bxb5+ 3.Kd2. The threat is now 4.Rg4 mate, and 3...Nh6 is met by 4.Be7 followed by Bg5 mate. So the bishop must return with 3...Bd7. So what difference has that made? We can now play 4.Nd5+ Kf5 5.Ne7+ and since d7 is blocked, Black must play 5...Kf4 (Ke6 allows Rg6 mate). Now 6.Ng6+ Kf5 7.Nh4+ is met by 7...Ke6! so it's time for a repeat performance: 6.Ke2! Bb5+ 7.c4! Bxc4+ 8.Kd2! Be6 and finally 9.Ng6+ Kf5 10.Nh4+ Kf4 11.Nd3 mate.