A long way...
Move 2 ... Qxg5 would be a better response for black.
The final position is a pretty mate, but the rest is somewhat sloppy for Black.
Qe1+ would win the other queen
7.Qd2+ Kb5 8.Qe2+ Nd3!
Makes things quite a bit more difficult for white. Then after 9.Qxd3 Kb6, it is white's knight to black's two passed pawns.