You’ve been left the registration number of the offending vehicle and details of an independent witness.
Follow this link which will provide you with details of the insurers of the at fault party:
http://www.askmid.com/askmidenquiry.aspx
Once you get the details of the insurer, I recommend calling them and advising them of what has happened plus the fact that you have independent witness who will place their insured vehicle at the scene of the accident. Pressure them into agreeing to fix your car.
This happened to me and the insurer folded after a week or so of me nagging them. I didn’t even have details of a witness, just some kind soul who left me details of the car that hit me but not their own details.
Alternatively for an easy life, let your own insurer fix it, pay your excess and hope that they make a full recovery from the third party.