The biggest problem is the time between deaths needs to be instantaneous. Take the tried and true example of Super Meat Boy, where respawn times do not exist.
Someone mentioned having a toggle for spacebar, or even just setting WASD to shoot. I think that would have been better. Even with the current setup, you didn't have to make my hands so close together, you could have set 'shoot' to an arrow key instead.
Turrets don't need to kill the player on contact. They already shoot stuff that kills you, there's no need to enhance the difficulty there.
Music was okay, not terrible. But maybe it would've been better if you just put ambient noise instead. The sound of the gun was good. Visuals are good. Interesting concept.