In one of the post game interviews Kramnik said he played well against the KID. He also said he wished he could play more of them.
He might be more well known for the Bayonet Attack, ECO 97, but he does have some strength in the Petrosian variation, ECO 92, where d5 is early followed by Bg5. There is his after game analysis on the 2014 London Classic win over Nakamura on YouTube.
I want to preface this by saying that, in general, studying the older masters is often easier and more beneficial for beginner and intermediate players. All of Kramnik, Karpov and Rubinstein were brilliantly strong players, but of the three, Rubinstein's games are the most accessible: his play is still crystal clear, but his opponents tend to defend with natural looking moves (rather than the near super-human, ultra defensive, Silicon assisted moves). By seeing how Rubinstein beats the natural looking moves, you can then see why the normal moves don't work and how the better but less obvious moves improve.
With that said, as far as modern players go, Kramnik is one of the better ones to study, and as a former world champ, there are many analysises and collections of his games.
Regardless of who you choose, Kramnik or a classic master, grab a collection of annotated games, preferably with more verbal explanations as opposed to endless variations, and study those. that seems far more profitable than randomly searching databases and zipping through a game at 2-3min a pop.
This is his most instructive game.
Kasparov was protesting the tournament in that game. He kept getting paired with men.
Nevertheless, a true masterpiece by Kramnik. Kasparov's ulterior motives should not get in the way of Kramnik's brilliancy!
Here's a game of him against Nakamura in the King's Indian.
The nice thing about this variation is that if black doesn't play 9. ... Ng4 white will play 10. Nd2! and you will reach the perfect piece placement white could possibly have against the king's indian.
(Nc3, Nd2, Be3 and Be2) usually white isn't able to reach this ideal setup.
Even if Ng4 is played which is the most critical variation against this line the position remains difficult.
Also you can be sure that your opponent won't be prepared against this line.
I've been interested in the idea of Bg5 in the KID, to provoke h6, when black will have to expose his king uncomfortably if he wants to carry on with his typical ...f5 play.
Holy moly, Bg5 to e3 to d2 all this to provoke h6 and then exploiting it with Qc1???
This some next level chess here. I think I might try this in my chess club. I'm so annoyed by all this kings indian players they always play the same stuff. Even if they are in a bad position they still have some cheap tricks at their disposal.
I think with a little bit of preparation this should be a good variation to learn because there is no deviation after d5 no exd4-stuff.
I am not an expert on Kramnik and his style changes but I am seeing a lot of queens pawn and reti openings from him as white in the last few years. Also the Catalan which is part of a repertoire like this. A good player to study for this type of game as white is Ulf Andersson. There's a good book about his games organized by theme called grandmaster chess strategy. His play is really clear and not too difficult to understand with the help of the annotations and explanations in this book. Another player to check out is Levon Aronian. He mostly plays queens pawn and reti as well. His style is more like that of Kramnik in it's agression and is more complicated than that of Andersson. The best thing about Aronian is that he plays with imagination like I have never seen. Check out his game vs Nepomniatchti in the 2017 Sinquefeild Cup. His game vs Svidler from the same tournament is very interesting as well. Hope this was helpful. I enjoy all 3 of these players for similar reasons and hope you will see what I do in their play!
"... Symmetrical English ... King's Indian ... Queen's Pawn Game ... Queen's Pawn Game ... King's Indian ... Ruy Lopez ... Giuoco Piano ... Giuoco Piano ... Ruy Lopez ... Ruy Lopez ... Giuoco Piano ... Giuoco Piano ... Sicilian ... Giuoco Piano ... Queen's Pawn Game ... Exchange Caro-Kann ... Queen's Pawn Game ... Fianchetto King's Indian without c4 ... Reti Opening ... Queen's Pawn Game ... Reti Opening ... Exchange Caro-Kann ... Dutch ... King's Indian ... Reti Opening ... Queen's Pawn Game ... English" - chessgames list of 2017 games played by Vladimir Kramnik as White (skipping those identified as Blitz or Rapid)
http://www.chessgames.com/perl/chess.pl?page=113&pid=12295
Grandmaster Chess Strategy by Jürgen Kaufeld & Guido Kern
https://web.archive.org/web/20140708093410/http://www.chesscafe.com/text/review812.pdf
Anyone who wants to learn from Kramnik games, my advise is to go for games in 90's , particularly 1992-1999, which i consider he performed the best with his positional chess, even later on, but many games from mentioned area contain lots of material for learning. also go for the games which he played against lower rated opponents (2500 and less) to see how small advatages are turned into winning position. he used to win Kasparov too in KID, his favourite weapon as black...
https://thechessworld.com/articles/general-information/top-10-chess-quotes-by-vladimir-quotes2/