If the dog owner has homeowners or renters insurance,they will pay your claim. Send her a letter demanding payment immediately or the info regarding her insurance. Sue her if she ignores you. I hope your dog is ok.
The owner of the aggressive doing should pay for the veterinarian bills incurred. If the owner fails to respond to your messages and/or refuses to turn the matter over to the homeowner's insurance carrier, you should file a small claim lawsuit against the owner of the aggressive dog.
It looks like your best path forward is to sue the owner of the other dog in small claims court.
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