how to win

great

6.Qh8# ... although if the board were reversed and that pawn were about to promote your moves are a little more valid.

First, the diagram is blasted upsidedown, even the FEN says it's upsidedown, would you mind correcting it since it is a bit confusing (my suggestion would be to change the black/white pieces to avoid screwing up any further notation).  To other posters - black is one move away from promoting in this position, ignore what the FEN says. 

Now the rest of my post isn't meant to nitpick or insult, it's meant to try to checkmate in the fewest amount of moves because I see posts like this as a contest.  First, there is a faster way to mate near the end of your idea, with 9. Qf7 actually forcing blacks king to run towards yours.  Further expanding on your idea, instead of playing 1. Qc5 Ka8 2. Qa5+,  you can mate faster with the tricky 1. Qc5 Ka8 2. Qc6! 

Now if 2... Kb8 3. Kb5 △ 4. Qa6+ Kb8 5. Kc6 Kc8 6. Qxb7 Kd8 7. Qd7# 

The trickier answer is 2... Ka7 3. Ka5! (3... b8=Q?! 4. Qa6#) b8=N! 4. Qc7+ Ka8 5. Kb6 Nd7+ 6. Qxd7 Kb8 7. Qb7#

Mate in 7 :D.  Can anyone beat that?

haha eric you love to comment about how great your own posts are