you got to play 17. Ne7+ straight away then Qh5 is mate!! or even 15 Bxc8 with the same
Before that, as soon as the h file is opened, go kg2 and Rh1 and you should checkmate
Earlier he should play 4... Qxe4+ picking up your h1 rook or the N. You counter with Qe2 and on Qxh1 play Ng6+ getting the h8 rook, but black can come at you hard with d6 and Bh3 which on the face of it gives black a winning position. I think!
No probs. But remember, the king is worth everything, so taking a queen rather than mating is a bit like taking a pawn rather than a queen!
After you played Qh6, go Rh1 next move. Why f4? You will mate him.
I think you could look at some standard mating attacks and tactics with great effect. All the best.
Like mxdplay4 said, (4. g3) is not good, leading to the King-Rook fork (4. ... Qxe4+ 5. ... Qxh1). Just lucky your opponent played poorly 
There are other tactics that you both missed (although reading it is rather dull, I'd recommend www.chesstactics.org for learning them).
Analysis would be greatly appreciated.