Some really good suggestions especially vertical kicking and Superman glide.
If you are swimming whole stroke predominantly maybe do some lengths of 'no kick' swimming. That is just focus on keeping your legs quiet allowing them to move with your hip/core rotation yet concentrate on keeping your feet close together.
Become hyper-aware of how your legs/feet behave as you swim. As you develop a rhythm add a snap to (one) downbeat and continue 'not' kicking. A few more strokes and maybe snap with the opposite downbeat leg. Mental notes.
Now perhaps two simultaneous snaps, rhythm, rhythm, snap snap. What do you feel? Be deliberate in your movements yet always use light pressure. As mentioned loose legs but keep "tone" (not tension) to your posture especially your core.
If your balance is great this will develop smoothly.
If you're working on breathing maybe try lengths of taking 3 - 5 strokes then sweetspot breathing and so on to maintain a rhythm without too much interruption or awkwardness.
Soon you'll incorporate the breathing exercise with the kicking focus and may discover the 'new' kick ( together with smooth hip/core rotation) will enable 'easier' breathing overall.
Play with your x/y coordinates. You may find as your kick develops, together with great balance, that you'll be able to raise your lead arm a bit also allowing for a smoother rotation to air.