At this point its kind of tough.
When you need something done, don't apologize and don't give anybody an out. Imagine if your boss called you on your day off and said "this is important, I need you to come in to work right now." You would probably be more likely to go in than if he said "I know you're probably really busy, but it would be great if you could come it to work today. I understand if you can't make it." Certainly don't thank him for not helping you...
You might email him back and tell him that you weren't able to find 3 other letter writers and you really need him to write you a (strong) letter outlining your research ability and that he is in the best position to do this. Also, let him know how important it is to you.
Good luck.