I'm not a lawyer or anything, but since the at-fault driver is not cooperating, you may want to get a police report of the accident.
Other than contacting a car accident lawyer in your area, it doesn't seem there's really much else you can do other than wait. If you do retain a lawyer, I think you should make sure you send a letter to the at-fault driver telling them you've done so. If many cases, at-fault drivers tend to cooperate once a lawyer gets involved.
Another option could be to go to your state's DMV website and fill out and file an SR-1 form (Proof of Financial Responsibility). Once an SR-1 form has been processed, the at-fault driver will have to contact the DMV and his insurance company. I'm not positive on this, but if there are no injuries from the accident your case could end up in small claims court.
I'm not an attorney and you need to remember auto accident insurance statutes are different in every state, but hopefully this link can provide you with some clarity:
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