Hi, Welcome to JustAnswer. I am a licensed veterinarian and I'm here to help you. Ouch! Poor Jasper! Amazingly, these broken and ripped out claws can become a major deal because very easily the toe can become infected and lead to a possible amputation.
The best thing would be to have your vet see him, clean and bandage his foot, and get him started on a systemic antibiotic. You are doing the correct thing for now. . .wrapping the foot to stop any major bleeding.