Is there is supposed to be more to it than the jump prototype demo?
Depleting your health doesn't do anything, and controlling the character is tricky. Not much else to do that I found.
For the wall jumps, consider coyote time. At the moment trying to move away as you jump often results in no jump. For the moment, don't give the player so much inertia. Try checking out the movement mechanics in Celeste; they talk about what they did and how they did it in various places. :)
After bouncing around and things, the character seemed to still have some basic inertia going on, so he would move a little every second. Purely velocity-driven movement works best in a space or highly physics-based game, but not a standard platformer. At the very least, you probably want to introduce damping into your velocity. :)
I'd be interested in hearing what your team learned during this jam, if you'd be willing to share. :)