I would get the details and talk to the car owner - tell her that you have witnesses that saw her hit your bike and that you tracked her down from the number plate. Imply that you got the plate details from the witnesses and get her to admit to everything - then it doesn't matter that you didn't have the plate details at the time. If she thinks she is sprung and admits liability and pays the costs to fix the bike then it is all good.
Unfortunately if she denies everything then it would be an absolute bastard trying to prove it was her and getting some money.
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