That sucks. When you go to any doctor, you generally need to sign something that says you're responsible for any payments that the insurance doesn't cover. Clearly, these payments are the responsibility of the insurance company, but that won't matter to the provider.
If I were in your shoes, I would be calling the insurance company on a daily basis until they paid the provider. I would also calmly be threatening to sue them and their client. If you are persistent, I suspect they will do the right thing.
You said you closed the claim, which I assume means you signed a form freeing them of any future liability. However, it seems you could make the case that you were deceived into thinking they had paid the bill, when, in fact, they only paid part of it.
Also, what about your insurance company? Do you have any sort of uninsured or underinsured coverage? If your insurance company is responsible for any payment, then THEY will contact the other driver on your behalf.