You've got 85% of both kicks -- solid buoyancy, solid trim, not dropping your knees at all.
What I'm seeing in the helicopter turn is that the back kick component is actually better than the frog kick, which makes me think your frog kick probably doesn't have enough ankle movement in it. My Fundies instructor told us to visualize the helicopter turn as one foot scooping the water to the other, and the second foot kicking it away. If you look at the video, your "kick away" part is pretty good, but your "scoop to" isn't scooping much.
In the back kick, I totally agree -- slow down, particularly the loading phase. Bring the feet closer together and extend them slowly.
For both kicks, reduce the amount of motion in the thighs and knees. (This will work with the frog kick, too. One of the effective things Natalie Gibb did for me during my sidemount class was to make me frog kick with a spool held between my knees. It drove me crazy, but once I figured it out, I had a LOT more power.)