How should black save himself?

Let's be honest. Black's position looks like crap. His rook on a8 is hanging and it doesn't look like there is much to do about it.

Black to play. Can you help find the way to make the position playable for black?

My best shot? It really does look hopeless for black.

I would guess that 3. Qd2 is met by something along the lines of Nxf4 Qxf4 Bh6. If Nxa8 there, then Bxf4 Qg3 Qa5+ Ke2 (Kd1 Qd2#, Qc3 Qxc3+ bxc3 Bf4 traps the knight) Qd2+ Kf3 h5 (threatening Bg4+) h3 Bf5 with huge threats.

Checked my engine, and it seems I was somewhat on the right path, other than some major move order inaccuracies and some problems with my h4 variation, as expected.

For example I was completely blundering 8. hxg5 Qxg5 Rxh6. But I won't give away the solution with engine lines.