It's tempting to lead another Spade to take out Dummy's last trump.

Before you do that look at Dummy again. East has a powerhouse (at least 19 points and six Spades for their bid). That could include ♣ A K in which case Dummy's Clubs can be set up.

The vital defence card is to play a Heart at trick two. You have to try and knock out Dummy's entry to the Clubs before they are set up!

You should bid 7♠ .

You can count four Aces, four Kings, and two Queens. Assuming the Spades behave you can ruff a Heart in hand for an 11th trick and Dummy's last trump then makes your 12th.

However Partner has shown at least 8 cards in the Majors.  If he has 4 Diamonds then they will split or you can cash ♦ A K then discard two on the ♣ A Q (because he will have a singleton ♣ K) and ruff the last Diamonds good. If he has three then you can still follow the same line as he only has two Clubs and you still get a discard.

If he has only two Diamonds then his 4♠ has shown 19 points. Two Aces = 8 and three Kings makes 17. You know he has ♦ A K and if he only has two Diamonds then he cannot have the ♦ J. Therefore his last two points will either be the ♥ Q (in which case you can discard two losing Diamonds on ♥ K Q and ruff the third) or the ♠ J and ♣ J (in which case you have to hope he has ♠ 10 so you can safely ruff two Diamonds

7♠ is making unless partner has precisely two Diamonds, both the Black Jacks, but is missing the ♠ 10. Even in that unlikely scenario a 3-3 Diamond Split still brings the Slam home.