This is a tough one.
First of all, let me say that you might find studying the actual King's Indian Defense useful.
1.d4 Nf6 2.c4 g6 3.Nc3 Bg7 4.e4 d6 5.Nf3 0-0 6.Be2 Nbd7 7.0-0 e5 8.dxe5 dxe5 is a direct transposition with the other side having an extra move.
See how black plays it with a tempo less, and you will really be on to something.
You have a position with little or no pawn play here.
Your main trump, as you mentioned is the square d5.
If you can lodge a piece there, it can't be chased away by a pawn.
Black can put a knight on d4, but c3 will always flick it away.
You need the maneuver Nc4 with the possibility of staying there (perhaps with a4) or running to d5 via e3.
First, you need a good move to defend e4 and prevent a queen trade. This is a lot harder to win without queens.
Here I am having a problem though.
1.Qe2 can be met by the messy Nd4, possibly.
1.c3 weakens d3 and allows Qd3. This looks a little pesky, but it really might not be all that bad for white.
For instance 1.c3 Qd3 2.Qa4 or 1.c3 Qd3 2.Re1 with Bf1 to follow if necessary.
The ideal formation for white is Qe2, pawn on c3, Nc4, and rook on d file.
Then, little by little you have to try to get more coverage of d5.
Maybe f3, to support e4 and free the bishop to move to f1 and possibly c4.
It's just a slow build up over the d5 square.
You can't really play for f4 or anything like that without taking a few positional risks in my opinion.
There is a similar position in the Ruy Lopez.
Winning from this position repeatedly, was Fischer's bread and butter.
You might want to find a few of his Ruy's for more ideas.
I'm sorry I can't be of more help, but this is the kind of position where I just know a few general positional ideas and just have to play it out and see.
There is no guarantee that white has an advantage with the King's Indian Attack. A lot of the positions are equal but imbalanced.
I'd say this is equal, but not without chances for white.
I hope this helps some.
I am just remembering another possibility, borrowed from Bobby.
Ndf1, Ne3, Nh4, and Nhf5 ... if black plays g6 the h6 square will be a nice square for a white bishop or knight. This leads to kingside possibilities.