write to DVLA and request it
https://www.gov.uk/request-information-from-dvlatrust me, when the court papers land on their mat they'll panic!
sorry just read the thread again - don't bother with the solicitor for the failing to stop bit - that is a criminal matter that police should follow up on. whether or not there is a conviction there will not change your case. Best to complain to the police that a crime was committed ( failing to stop after an accident) and they did not follow up.
The issue you need to pursue is the costs of repairing the car. so
get the keepers details - send the a quote for the repair and 30 day payment terms.
then if they do not pay, send them a demand
then if they do not pay, go to small claims - you need to be seen to act reasonably here in pursuing the debt but you should persist - it may take 6 months but either this will go to court or it will get said person back to the table to settle via their insurers
also have you tried speaking to anyone near where the incident occurred to see if they have any CCTV covering the area - would be fantastic if you could prove the vehicle was in the area at the time