Consider opponent's problems

Second hand, I open 1H and left-hand opponent overcalls 2S.  That should be a weak 6-card suit and I find myself in 3NT.  I get the lead of a small spade. Dummy comes down and I see that partner has the Queen of Spades as well, but no entries.  How would you play?

Me, I played it badly: I reasoned that I probably had only one entry to dummy. So, I cashed my King of spades, took my five hearts and then broached diamonds. Of course, I had only 7 tricks and after giving me the queen of spades, they took the rest. There are two lines that win here:

1. After cashing the king of spades, lead towards the king of clubs.  This wins if North has the Ace of clubs, as is likely. He must have some points somewhere.
2. Don't cash the king of spades. Instead, start working on diamonds.  The opponents will lead spades for me. What else are they going to lead?

Obviously, the second line is best but it was hard to see at the table. I needed to stop thinking about getting to dummy to cash the Queen spade and realize that any other suit that the opponents lead will lead to my ninth trick in that suit.