Super original and weird gameplay. I love it! The physics are balanced such that you could never just do one thing over and over to guarantee success. Sometimes you have to make a really big sword swing by building up momentum. Sometimes you have to go for smaller sword taps. Sometimes you have the sword lined up just right to drive a high-ranking enemy into a corner. Great take on the theme!
My main criticism is that the sword-width powerup is super duper busted. I think what you were going for is that you need to upgrade your sword's length in order to get enough damage to not get outclassed by enemies, but the catch is that your sword gets harder and harder to control which increases the difficulty. But sword-width also increases damage and doesn't have that downside. Once I got my sword wide enough, I basically had an invincible shield around me at all times and I cruised through to round 11 with no problem.
So I decided to do one more run, this time without using the sword-width powerup. It definitely got more unwieldy as I had crowds of enemies in the middle of the arena and no easy way to rotate myself to get the sword towards them. But eventually I figured it out and learned some new tricks as well. If you can jam the sword between two enemies at full health, you can circle around them and twist them to death pretty quickly. You can also push the sword towards the wall at an angle and sweep across the room horizontally. It did still get really easy by the end though. So maybe it's not the sword-width powerup that's at fault. Maybe the game just doesn't scale in difficulty fast enough in the later rounds.
Anyway, I had a ton of fun with this one. Great work!