In addition to flexibility, it also takes some strength - I think the muscle groups are the hip abductors and the oblique abdominals.
One way to build this strength is to hang on to a stretching bar or counter and to slowly lift the leg into the extended kick position - as high as you can - and hold it there for as long as you can.
The supporting foot should be pointing the HEEL toward the target, and try to keep your body in a straight line behind the kick - don't "checkmark" your body.
Also, just keep pushing yourself - keep striving for high kicks (being sure to maintain good form - it can be dangerous to do so with sloppy form).
The more high kicking you attempt, the more you build those muscles you need for high kicking.
While I agree high kicks usually won't have a place for self-defense on the street (usually), they sure are FUN during free sparring.
AND - if you are comfortable throwing head-high kicks, then solar-plexus or other mid-range kicks can be executed with even more confidence and ease.
By the way, the muscle groups you work to achieve a head-high side kick should also enable you to do head-high roundhouse and hook kicks.
Good luck!