Is there a leash law in the area where this occurred? If so, then legally I don't think you owe the dog's owner anything. Assuming that there is a leash law in your area, and there was no leash on the dog, and you were driving within the speeding limit, the woman has no right to ask you to help pay the bill.
Many years ago, my dad was walking my dog (it was good for my dad to get out and walk; so we'd encourage him to walk my dog, a 49-lb. spaniel), he lost control of the leash and my dog trotted into the street and was hit by a car. The car driver stopped immediately and asked if the dog was alright (my dog had squealed and run away, thankfully off the street). I came out and found my dog, he appeared nervous and shaken; I leashed him and brought him immediately to the local vet hospital. We were very lucky; the driver had been going within the 25-mph speed limit and just clipped my dog on the head; my dog lived another 14 years. It never occurred to me or my mother or father to demand that the car driver to pay for the vet bill; because we know he was not at fault. (that was the last time my dad took my dog out on a leash; he just was not strong enough anymore to control a rambunctious young dog).
You might want to ask a lawyer, or perhaps the local animal control officer or the police, whether you are legally obligated to pay this bill; I personally don't think so.