To elaborate a bit on
I thought it would be checkmate, because anywhere the king goes he'll
This will never happen, though. A player must never oppose his king to attack, so he just has not a single move he can make. If it was ok to play an illegal move (moving into check), then he could just make any other illegal move at well, like capturing your king with his king, or capturing your queen - there is really no difference.
And as he's neither attacked, nor the game can advance any further, the game is drawn per the rules, which seems a natural choice.
Note that other chess variants (also Chinese Chess) might treat stalemate differently.