To wall jump you need a small running start. On a single flat wall you can only jump up once, but with 2 walls you should be able to go up a long way, however you will need to change directions midair.
Added a few more levels to the end. Fixed the terrain moving slightly after the non-terrain objects. For some reason the code that runs when the player dies was disabled. Fixed. For the record, I managed to speed-run the whole thing without dying once, though I didn't time it.