Keep him in the fridge. Try not to bother him too much, as it will stress him out to have the light going on/off/on/off and you looming over him. Use clean, dechlorinated water. Offer food, but don't be surprised if he doesn't eat.
And just wait. Axolotls are masters at healing, and often surprise their owners with what they will pull through.
Do keep an eye out for fungus. It looks like cotton wool or fuzzy growth on the wounds. If you do notice fungus, you'll need to start salt baths. Don't do salt baths unless you see fungus.