Could black win this game?

first play 1... Nf6+, and then you must find a way to win manouvering the rooks and knight.
Bring the white king to the d file, to check with a rook and win the queen.
Or the knight can make a fork on the way, in the squares c5,e3 and f2.

You must win this position, the white king is terrible exposed. 

Nf6+ is a natural attacking check. The knight can participate in a strong attack. Black is completely in control.

 For example, 1...Nf6+ 2 Kg3 Rg5+ 3 Kh3 (Kf3 Ref5+ Ke3 Ng4+ and Black wins) Rh5+
4 Kg3 Reg5+ 5 Kf3 Rf5+ 6 Kg3 Ne4+ 7 Kg2 Rf2+ 8 Kg1 Rg5+ 9 Kh1 Rd2 10 Qe1 Rd1 and Black wins.

...  Rg5+; Kf4 Rf5+; Ke4 Nc5+ wins rook.

Of course there are many variations for white but it's just an example of pushing the king around on squares for forks, or in one line with the queen or so on. Getting the knight into play is a sound plan, just keep checking. Black is winning