I understand that if you have reasonable cause for requiring details of ownership of another vehicle, then you can apply to the DVLA for keeper details.
https://www.gov.uk/request-information-from-dvlaOnly you can say whether you feel it is worth pursuing ie your speed relative to the other vehicle, the relevant positions of both your vehicles on the road when the impact took place, and whether the dash cam clearly shows the other vehicle at fault.
How amenable the other driver is to coughing up for your mirror casing is another matter which needs to considered, along with where the other vehicle is registered. Is it something that could be resolved by letter if not local to you?
All in all, unless local to you, and with concrete evidence on your dash cam, I would say it is something you are going to have to grin and bear. Its going to probably cost you more in faffing about than the monetary value of a new mirror casing / painting / fitting.