A really neat concept, using your own death to get further into the game! :)
I didn't really see the need for the jump cooldown though, and I think allowing the player to control movement while jumping would make it a lot smoother (I find that taking away control from the player makes it feel like a death was the game's fault rather than the player's, which is far more frustrating than if it's the other way around). Also, is the attack used for anything? I tried attacking the enemy but every time I got close enough I died and respawned at the beginning (and since red bars are usually associated with HP, it took me a while to realize it was related to the attack cooldown).
Otherwise, the pixel art, animations and the setting are really nice, the menus and UI were clean and not too complicated, the dialogue telling you how many times you'd died was a nice touch. Reminded me of the monument in Majula in Dark Souls 2 (R.I.P).