doing static flexibility exercises, for example, touching your chest to your leg will contribute to overall flexibility, but not necessarily kicking flexibility.
The best way to gain this type of flexibility is with dynamic stretching exercises.
first exercise: swing straight leg to the front as high as you can. Hold your outstretched arms out in front for a target.
second exercise : swing your leg out to the side stretching the inner thigh, again having your outstretched arm held out for a target.
third exercise : looking over your right shoulder swing your right leg to the rear as high as you can, torso will be inclined forward on this stretch. (And then repeat for left side)
IMPORTANT: warm up before doing these stretches. Many people think of stretching as warming up, it's not. Run in place, do a stationary bike, jump rope, whatever to get a light sweat going and heart pumping a little, then perform these stretches. I do 3 sets of 10 of each stretch. Also don't try to go too far too fast. You may throw something out you'll want later. So progress slowly.
Then after your workout do the old fashioned static stretching as your cool down.
Another tip. When kicking, practice slowly to bring more muscle memory into the kick instead of just relying on momentum.