I've searched everywhere, however, I couldn't find what are ideas in KIA. I know about Neil McDonal "The King's Indian Attack: Move by Move" book. This book covers KIA against different openings. However, I'm struggling when my opponent stops playing opening moves, I find myself in situation where I don't know how to continue.
What is general idea in KIA? Launch kingside attack, queenside attack? Or something else?
The best way to learn the KIA is to study master games. White attacks on the kingside, and black attack on he queenside. Study Fischer's KIA games as well as the KIA games of others.
From the books The Ultimate King's Indian Attack by Dunnington and Modern King's Indian Attack by Hall, there can be attacks on either side or a center conflict. Since I play the Sicilian Dragon, I usually play on the queenside. If the opponent plays an early d5, I transpose the game into a Catalan and pressure the d5 pawn. If my opponent does not fianchetto either Bishop, I overprotect e5 and play for a kingside attack.
Where you attack depends upon the opponent's pawn structure. As suggested by Jimmy360, most common Ideas can be learnt by playing over games featuring the KIA and the KID. See ideas for KIA.
Dynamism is at the heart of King's Indian attack (like many other openings). You cannot be fixed on always going for the kingside or alway going for the queenside. You have to be flexible with respect to all options including kingside, queenside, both sides or center conflict. Also bear in mind that it is a modern opening after all. So, regarding center and general plans, it shares traits of many other modern openings (see chapter 6 of Fine's book "The ideas behind chess openings" for instance). But, beware: the theory of modern openings have changed from its inception (esp. with respect to importance of center).
In some variations, you can strike back at the center in better circumstances than the opening (timing is important for this). Sometimes, you can even push the e4 pawn to e5 (this opportunity doesn't come often. This push will simply loss the pawn (and center dreams) more often). When the center is kind-of fixed, the typical pawn-structure advises white to go for a kingside attack, and black to go for a queenside attack.
See section C of exeterchessclub's openings page for a short overview. For more detailed one, read chapter 7 of "Winning chess openings" (extra read: 1st para. of chapter 5 of the same book).
