I don't think you should pay anything to be honest. In fact I think they should pay the vet bills for your dog as they kicked your dog and caused the injury.
Legally the police will do nothing for a dog on dog attack, so legally you are not in the wrong.
Did they put their dog to sleep on order of the vet? Or maybe they just could not be bothered to pay the vet bill. Sounds like they did not care about their dog very much and could well have been about to put it to sleep anyway (or get rid of it). A simple puncture wound like you describe would not need much treatment, usually only antibiotics.
Seems to me like they are trying to pull a fast one. Are they demanding that you pay the money to them?
I definitely would not pay anything.
That must have been a really hard kick to break a few ribs. She is really lucky that your dog did not turn on her (if it was my dog she would have got more than a mouth full off me

).
Hope your dog is feeling better soon.
Sue