Sort:  

Ok I also think this is the only line that can give a clear win. Ideas of Bf4 to trap the knight on f1 seems too slow and will allow black pieces to activate the rook... So:

1.Rxh5, gxh5
2.Bh6, Nf1
3.c5 dxc5
4.bxc5 and black cannot stop d6 d7 d8

3... Ng3
4.c6 bxc6
5.dxc6 Nxf5
6.b5 and now the a pawn will become the hero

Edit: I just noticed black can do Nxh6 and Kg7 to stop the pawn. Maybe we should keep e4 somewhere hehe

I will have time to reply later ... but indeed you should consider how game will continue after your 3. ... Ng3.

1.Rxh5, gxh5
2.Bh6, Nf1
3.e4, ......
After,
3.e4. Ng3 (Ng3 is a total waste of time by black. the black Night has no game left)

OR
3.e4, b6
4.b5, bxa5
5.bxa6, a4
6.a7, ......
(white queens first and wins the game)
IF blacks responds {4}....., axb5. White proceeds to queen with {5} a6, ..

OR
3.e4, c6
4.dxc6, bxc6
5.b5, cxb5
6.cxb5, axb5
7.a6, ...
(White goes on to win the game again)

  1. Rxh5 gxh5
  2. Bh6 Nf1
  3. e4
    Now, as you wrote, pawn moves like 3. ... b6 or 3. ... c6 are just suicide.
    Therefore the only moves which don't lose immediately are knight moves or 3. ... h4 (3. e4 is only a transposition of moves: in case of 3. c5 black answers with 3. ... Ng3, then 4. e4 is necessary).

But want I want to know from you is how exactly white wins after:

  1. Rxh5 gxh5
  2. Bh6 Nf1
  3. e4 Ng3
    How will you continue as white?

After black plays 4.Ng3
5.c5,dxc5
6.bxc5,Nh1 or Nf1
7.d6,cxd6
8.cxd6,Nf2+
9.Qd2,Nxh3
10.d7, black resigns.
(white queens with d8 and check mates black.)

  1. Rxh5 gxh5
  2. Bh6 Nf1
  3. e4 Ng3
  4. c5 h4 (If black takes the pawn it always ends fast.)
    How does white continue? (By the way neither 3. e4 nor 3. c5 are the fastest winning moves here, but I accept them as long as you win in the end. :) )

Maybe I should add why I am so persistent to play all moves until the bitter end: you should be aware that chess friends of many different levels are watching here. Not for all of them it is so obvious how white can win at all, if black just waits and just does nothing ... :)