The terms of your insurance policy require you to notify your insurers of any accident even if you don't intend to claim on that policy. The other driver's policy will have a similar condition. So it would be best to contact your insurers because, even if you don't, the other driver's insurance company will pass on the information anyway.
Contact the other driver and ask her to tell you, as soon as possible, what she's been quoted for repairing the damage. If you're happy to pay from your own pocket, pay her immediately (ensuring that you get a receipt for your payment, stating that it's in full and final settlement of any claim resulting from the accident).
Be prepared for a nasty shock when you get the quote. I've only ever been involved in one 'fault' accident. It looked as if the damage to the other driver's vehicle would cost well under £100 to fix, so I told him that I'd pay him directly. But he got three quotes, with the cheapest being about £800, so I decided to claim on my insurance!
Chris