Your daughter can pay £4 to get hold of the other driver's insurance details:
https://www.mib.org.uk/check-insurance-details/check-a-vehicle-not-at-the-roadside-after-an-accident/
Then she can ask her own insurer whether they'll assist in claiming against that insurance. (If not, she could try contacting the other driver's insurer directly).
Alternatively she can bypass the insurance system altogether and threaten to sue the other driver (following through with court action if required). That would then leave him to decide whether or not to claim on his insurance. (If he didn't he'd have to pay your daughter from his own pocket). Assuming that the driver was the registered keeper of the vehicle she can get his name and address by submitting form V888 to DVLA:
https://www.gov.uk/request-information-from-dvla
She'd then need to send a formal written demand to his address (using Royal Mail's 'Signed For' service), stating the amount demanded, the reason for the demand and that legal action would follow if payment wasn't received within a specified period (of, say, 21 days). If he failed to pay up she could then take court action against him to recover the money using the Government's online claims system:
https://www.moneyclaim.gov.uk/web/mcol/welcome