We had an accident that was entirely the fault of the other driver. He stepped up and admitted he was not paying attention and that he ran the red light.
OUR insurance company immediately supplied a rental car and said that they would be in touch with the other driver's insurance company. A week later, the other insurance company contacted us and said that they would be taking care of the rental charges and would be getting back to us about the damage to our vehicle.
One month later, we were notified that they were totaling our Jaguar and that they would be issuing us an agreed upon amount check for it's value.
Many states have uninsured motorist coverage but in this state it ONLY covers your injuries and does not cover repair of your vehicle. Since the other vehicle was insured, they will be responsible for the damage to your vehicle.
Because the other driver did not have a license, and was drunk, you do have excellent legal grounds for a civil lawsuit against the owner of the vehicle. I'm not saying that you should sue but it is something to keep in mind while you are negotiating a settlement. Good luck and the most important things is that you are all OK.