Car insurance, as the name suggests, is purchased for vehicles, not individuals. Within reason, insurance companies will extend the coverage to anyone who uses the car with the owner's permission. However, if the owner loans the car to a person without a driving license, the insurance company is not obligated to pay for the costs if an accident results. Furthermore, loaning your car to an unlicensed driver is a serious offense and could result in the loss of your insurance.
In state which have PIP, or Personal Injury Protection, the insurance plan belonging to the vehicle is the first one to pay for medical damages. Under PIP law, your own policy would pay for your injuries immediately, and then your insurance would settle with the other insurance company after the fact. This is meant to make certain that insured drivers have access to insured care immediately after an accident, whether the at fault driver had coverage or not.
File the claim. Contact your insurance company, state the facts, and you may find out that your insurer will be happy to handle the details of getting the claim resolved. As long as you are not at fault in the accident, you are entitled to repairs and medical care from the other drivers insurance.
If you have serious problems getting a claim settled, consider contacting an attorney. Getting a lawyer involved should be considered a last alternative if nothing else works to settle the claim. Insurance companies have lawyers as well, and bringing in the legal guns does not guarantee you will win, but it does guarantee you will not win the full settlement because of the fees that must be paid for legal counsel.