Kicking has four parts to it: flexibility, technique, focus and ab's.
For the flexibility, I have found PNF stretching to be quite beneficial. This is a form of stretching that uses periodic resistence/contraction followed by relaxation to achieve a deeper strectch and excellent long term results (here is a reasonable Youtube example). Of course flexibility is only part of the story and by itself doesn't get your foot up there, being flexible simply ensures that you have the minimum resistence to getting your foot up high.
For technique: concentrate on breaking down the kick into its component parts. Practice it slowly. First raise your knee, hold it, extend the leg, hold it, lower the leg but keep your knee up, hold it, then lower your leg back to where you started. These positions should all be held for several seconds each (holding them longer can help build tendon strength).
Breaking the kick down into its component parts helps you master the mechanics of it, and helps to prevent sloppy technique that is little more than a quick sideways leg raise rather than a good high sidekick. The groin adductor muscles which are the main muscles involved in getting the height (angle) on this kick are not very large, especially when compared to the total size of your thigh, you can work those muscles at the gym* but strengthening them is only going to get you part way there; what is more important is the speed and technique used. Once you have the kick at the desired angle your quads and glutes take over and extend the leg - those are easy to strengthen, but concentrate on them last (accuracy and technique is more important than brute strength at this stage).
Focus is very important. Your mind (focus) needs to be just beyond the edge of your foot (which should be through the other side of the target). Your focus needs to be impeccable, but it is hard to achieve when battling poor technique and/or tight/weak muscles. If you cannot kick at a height that satisfies you, then back off a little and perfect your kick at a lower level, then gradually work your way up in height. Don't be a glutton for instant gratification, this sort of thing can take a lot of time.
Finally, I cannot understate the amount of importance that ab's play in your kicking, even side kicks. Without strong abdominals your kicks will always be weak. They are a crucial support muscle used during the kick, without them you may as well just swing your leg around in a haphazard way.
*use the machine where you sit in it and spread your legs apart sideways (it will usually be situated near its counterpart where you squeeze your legs in together, which is the opposite group of muscles).