Bear in mind if you do go through your insurance, your premium will rise next year, unless you have a protected No Claims Bonus.
As veryoldbear says, it depends on the other party. She may welcome a quick fix of you offering to pay for the scratch repair. If it's more than a scratch, if there's any dent involved, then she may insist on a full bodyshop repair.
Last year I had a 17 year old back into my car at Sainsbury's. He really didn't want to ruin his insurance rating, so I got two quotes for the repair, he paid the for the lower one (£160). Both parties were happy.
At the risk of sounding condescending, well done for doing the right thing and leaving your details on the other car.
My 17 year old son did the same thing to a visiting dignitary in the school car park. She was furious and threatened all sorts but it really was a minor scratch and her husband later called to say send us £20 for a bottle of T Cut and I'll sort it. Son sent a bouquet of flowers as well and everyone was happy!