Who does the fence belong to? Was her dig's nose "trespassing" across the property line? The owner of the fence may have an obligation to maintain the fence and he/she is negligent in allowing a hold to remain, that could be mean he/she is legally and financially responsible for this incident. Homeowners insurance will likely pay a claim - file it and let the insurance companies duke it out. The first step, however, should always be to discuss it with the neighbors and see if they accept responsibility or not.
Statements made in response to questions on this site are for informational purposes only and do not constitute an attorney-client relationship.
There are two issues to be addressed; ownership of the fence - with the collateral issue of knowledge of the hole in it - and which dog's nose was in the other's yard. It sounds as if the fence is your fiancee's, meaning its maintenance is her responsibility. The source of the hole and the length of time it has been there may be issues.
Whether her dog stuck its nose into the other dog's yard while "sniffing by the fence" or the other dog, as you state, went through the hole to bite your fiancee's dog in its yard will be a point of contention. It appears you were the only witness.
Your fiancee will need to document all damages. She should have filed a report with the police and/or animal control at the time.
If the neighbors are not willing to work with her regarding the vet's bill, she may have to send a demand letter and then file an action in small claims court.
The foregoing is for general information purposes and does not establish an attorney-client relationship.