If you play the Pirc/Modern, 2...d6 or 2...g6, allowing 3. e4, will work fine.
If you play the Classical French, 2...e6 3. e4 d5 will work, although you might want to look at 3. Bg5 and other third move alternatives.
If you don't play any of those openings, I don't see a great alternative to 2...d5. 2...c5 is not terrible but it also leads to a specific sort of opening by transposition (Schmidt Benoni) which I doubt you play if you're telling me that you want to play the KID.
Therefore you may still want to play 2...d5. It's not very KID-esque but sometimes you have to do what you have to do, and I think black's perfectly fine.
I have been playing Nf6 against every 1. d4 game as black and been trying to practice my King's Indian Defense. I think on move 2 the typical ideas for white are bringing the dark square bishop out, or playing c4, followed by Nc3. As an amateur though I play against a lot of amateur players that don't play the most common and professional moves, and I think a common amateur player move here is 2. Nc3, blocking the c pawn.
When I try to set up the usual KID structure against this move I think it actually becomes a Pirc Defense, especially if they get in e4 immediately to justify the early Nc3 move. There's a way to stop e4 though by playing 2. d5, and thus it enters a "Queen's Pawn game". If I want to play the KID should I know some of these other defenses too? I was hoping I could just stick to the Indian defenses against 1. d4 for now but it seems like some moves just completely avoid them.