I am unsure how long you have practiced, so take these comments with a few qualifications if I am going over stuff you already know.
Generally, a kick requires flexibility...but also muscle strength. Flexibility also doesn't mean just in long stretches. For instance, just becuase one can do the splits doesn't mean this translates into a good side kick.
Flexibility in the hip joints and especially in the supporting leg is a requisite. The supporting leg, in fact, requires the brunt of necessary flex in order to move the hips over this support to throw a high kick.
Therefore, my suggestion is that in the normal course of just practicing these things in a class you will obtain the requisite strenght to hold the kick.
Work the kicks you want to increase height in slowly while maintaining contact with a wall is a good start and can be done at home. Try to keep good mechanics and first go for mid-height and hold the technique, withdrawing into the same chamber that you did to throw the technique. Biggest point here is to make sure that you "push" the hip into the kick.
Then raise the level of the kick and do it again.