If the other driver's insurance refuses to pay for your car repairs then file a claim with YOUR insurance company against your uninsured motorist coverage. They will pay for your repairs then sue the other driver for repayment.
If you don't have uninsured motorist coverage then get a quote for repairs, and file a lawsuit against the other driver.
EDIT: To answer your follow up question in the comment: If I were you and this happened to me with liability only coverage, here's how I would proceed:
1. I would call my insurance company and upgrade my coverage to include uninsured motorist coverage, as well as personal injury protection. I will never be without those coverage regardless of the fact that they aren't legally required.
2. I would take the car to at least 1 body repair shop (2 or 3 is better) and ask for a free estimate. Explain the situation and that you just need a rough estimate for insurance purposes, not necessarily a detailed quote. They will give you a dollar figure to work with which is at least close to what it would cost to undo the damage.
3. If the amount is less than the limit for small claims court in your area (where the accident occurred) then go to your local courthouse and file a small claims lawsuit against the driver. Most places have a $5000 limit for small claims court, but google the court system in your county to verify. Filing is easy, its just a few forms and a nominal fee.They will give you a date and time for the hearing and a packet of paperwork which you must "serve" to the defendant. You can serve them yourself or you can hire a process server to track them down and get the paperwork to them. If they don't show up for the hearing you'll have to prove to the judge that you served them, and if you didn't then your case will be dropped. If you have proof that you served them, then you'll win a judgement against them by default.
4. Usually once you serve them and they realize you're actually taking legal action, they'll step up and answer the phone calls of insurance and try to work with you to work out a settlement.
5. If they don't settle or get their insurance company to settle for them, then you get a judgement against them. Good luck trying to collect though. Start reading up on the process in your state, its good info to learn anyway even if you never get a dime from them for this accident.
6. Face the reality that you will likely never get paid. You chose to drive without uninsured motorist coverage and therefore you chose to take this risk (see step 1). You can either live with the damage, pay out of pocket to have it fixed, or buy another car. Either way this is an expensive lesson about knowing your insurance coverage and protecting yourself even when someone else should have had insurance to protect you.