If you were injured enough to need medical attention, then it is best to get a personal injury attorney. If it is just property damage, then it really is easiest to turn it over to your insurance company. There is no way you can force a settlement with the other side without filing a lawsuit. But your insurance company has other ways to get paid back and recover your deductible if you end up having to pay one.
You should have exchanged insurance information with the other person. Call your insurance company and report the accident. If you were injured get medical attention, take pictures of the damage, don't talk to the other side's insurance company before you hire a lawyer to help you. If there is only property damage, your insurance company will help you handle it.
The case depends on who was negligent. If 3 lawyers have looked at the case there is obviously a problem. You can talk to as many lawyers as will listen to you. Find a lawyer who is board certified in personal injury trial law. That lawyer will be able to give you an expert opinion.
Since the other driver has no insurance, there is no money to get, especially if the driver goes to jail. You cannot expect a lawyer to sue for a percent of money received (a contingent fee) if there is no money to get. To get a lawyer to take the case, you will have to pay an hourly fee for work done on the case.
You can expect that the offer will be vary low. I cannot determine what your case may be worth, from a question posted here. I would consult with a local personal injury attorney to go over the details.
You are just going to have to get a lawyer. There is nothing else you can do to make them respond. You might have some trouble finding someone who is willing to get involved at this point, but keep looking, you will find someone.
Send a certified letter saying you do not have the documents if the subpoena is only for the documents. BUT IF THE SUBPOENA ALSO SAYS YOU MUST APPEAR IN COURT OR FOR A DEPOSITION, THEN YOU MUST GO IN PERSON.
How bizarre!! You may have a claim, but the question will be whether there are any damages in addition to the property damage. It does not sound like there are any. There are always two questions in a lawsuit: are they responsible, and if so, how much were the damages.