Now, suppose your neighbor removed her fence entirely... How would you keep your dogs in your yard?
Your neighbor is responsible for keeping her dogs in her yard - that's it. She has no responsibility for your dogs - that's your job. If your dogs have crossed into her property and injured her pets, then yes, you are on the hook for the vet bills. Furthermore, if your dogs are prone to fighting with other dogs, then erect YOUR OWN fence to control your dogs.
So, yes, you are probably responsible for vet bills, if the neighbors dogs were injured on the neighbor's property.
If the neighbor can PROVE that your dogs damaged her fence, causing it to fail, then you are responsible for repairing the damage caused by your dogs.
Otherwise, there is shared responsibility in what has happened. If her dogs escaped the yard and were injured on your property, then she has some fault for not confining her dogs to her property.
However, from your attitude and her reaction, it sounds as though you expect her to be responsible for keeping your dogs out of her yard. She is not. You are responsible for keeping YOUR dogs in YOUR yard. If you were depending on someone else's fence to do that, then shame on you: built your own fence. Expensive, yes - but that's part of the cost (and consequences) of having dog(s).