You should have called the police and got a report. They assign fault on the spot if it should be assigned. That fault determines whose insurer will pay for damages automatically. You go to the other persons insurance with a plice report saying that their client was a fault and damaged your car, and they get your car fixed.
If you don't get a report you can get yourself into trouble because the insurance of the party at fault can and might try to deny the fact that their client damaged your car. It has happened to several family members of mine. This is why I always get a report, then they pay no questions asked. The insurance simply won't deny their client's responsibility if the police report says they are at fault.
