Property damage claims can sometimes be very frustrating if the at-fault driver's insurance company does not admit liability right away. The insurance company needs to conduct its own investigation of the accident, which can often times take weeks to complete. Once liability has been accepted, the insurance company will pay for the repairs or even total the vehicle if the cost of the repairs exceeds the current value of the vehicle. They will also provide a rental vehicle while the vehicle is being repaired. You have a right to have the vehicle repaired at any place you choose.
If the insurance company refuses to accept liability, another option is to have the vehicle repaired through your own insurance, provided you have collision coverage on your policy. You will need to pay the deductible, but this will be reimbursed to you once the other driver's insurance carrier is found to be at fault.
If your vehicle needed to be towed from the accident, you have a duty to mitigate the storage fees and get the vehicle out, as the insurance company will not be responsible for these excessive charges.