I like to play a strategical KID, yet I still get a decent share of Kingside attacks. It seems that a lot of players are attracted to the KID via sac, sac, mate del Plata, but not me.
If you read Gallagher's books, he seems to want this sort of activty in all lines, and so hates the Exchange and Fianchetto variations. These do not worry me at all. OK, maybe tiny advantage for white, or best hopes of dull equality for Black. I will not let my mind be bothered by that theortical issue, and get on with the practical game. At my level, there are enough mistakes for things to be interesting.
I like the Nbd7 Classical KID. I've studied Nc6, but never tried to really memorise or play the variations. I've dabbled with Na6 (based on Gallagher's book). I see the logic for the argument for why Na6 is meant to better than Nbd7, but don't see it in practice.
I like that there's not been a repertoire book (well not one I've found, except maybe for DWs) on Nbd7 as it forced me to do my own study. I studied it by, well initially not really knowing what I was doing, and eventually by fishing out games, especially annotated ones, with Chessbase from GMs/IMs who use it. There are plenty, and they still continue to. So I would look at what repertoires they used. There seemed to be a number of options, which is something I like to see when playing an opening.
I like the Yelena Dembo DW coverage of Nbd7. I saw these lines appearing from my Chessbase repertoire scans and coverage on this site, and it was nice to see a bit more explanation behind it. I've got her anti-KID book too, so I guess she's got me covered now
Another useful facet of Nbd7 is that it allows me to enter the KID via 1. ...d6, and avoid a whole host of tricky White systems by playing Nbd7 early, and only playing Nf6 after White has gone Nf3. No Samisch, no 4PA, no Averbach, no Nge2 systems etc.
The downside, if you can call it that, is that White can go 2. e4. Now the hope here is that a d4 player is far less prepared and experienced in this than an e4 player. Not always true. I seem to get more d4 players going for e4 lately. But Black doesn't need to play a Pirc/modern here. One d4 player said he'd spent the morning preparing a 150 attack against my Pirc, but after 1. d4 d6 2. e4 Nf6 3. Nc3 c6, he was totally out of water
But I digress. Sometimes I think the KID is just positionally busted and you need to get some tactics (which somehow seem to materialise by magic

) and so how can you play it strategically? Well this seems to work out in practice. White doesn't seem to have perfected an antidote so far.
So I get mostly Nbd7 classicals, Fianchettos (I like to play it old school style, but there are options), and for the wimp outs, Londons and e3 systems (thank you for the equality). Playing this move order might not suit your repertoire, but I think it's fine even if you have hopes of a GM title