
If a dog bites a person in the park whilst in the care of a dog sitter or dog walker then it is the dog walker/sitters responsibility in law and that is why professional sitters & walker have insurance.
If the cat had maimed one of the dogs she too would be liable since both cat and dogs were in her care. No-one afterall pays for their dog to be looked after only to find they are injured by the carers other animals. That is also why sitters/walkers need insurance. Though many pet insurers simply wont offer third party liability on cats insurance anyway so if the cat had injured your dogs she would be unlikely to be covered under her cat insurance - only under a pet sitter insurance policy and even then its unlikely to be covered.
It cannot be any different just because the incident occurred in her own home with her own cat especially as she had been warned not to allow the dogs to be in contact with a cat and was negligent in her responsibilities in ensuring this did not happen. (ie. by not staying with the dogs she was paid to care for or leaving them only in the care of another informed and responsible adult).
Lastly, who witnessed the dog/cat fight that is not an interested party? Since the owner of the dogs wasn't there and the owner of the cat is biased. There is no proof that the dog ever had a fight with the cat at all (and if that were proven it would have to be shown exactly which dog caused the injury or that they BOTH jointly caused injury). As far as you know the injured the cat it could quite easily have gone out that night, got into a fight with another animal and the owner made the whole thing up to try to get you to pay out for it. No-one is liable unless there is proof of which animal did what to whom, why and when.
Unless you want to pay out as an act of unnecessary kindness & charity - refuse the money for the vet bills or anything. Do not accept liability as you cannot know for certain that your dogs did anything wrong at all - you do NOT know that they did - you didn't see it - you have no independent witness and nothing but hearsay from an interested party that the fracas ever occurred in the way you were informed. (Pet insurance companies specifically inform pet owners not to accept any liability unless they were a witness). If she pursues it she will get nowhere - in fact if your dogs suffered any injury, behavioural changes, further fear of cats or stress from the incident quite frankly she would be liable no matter what injury the cat suffered. The only time this is not the case is if it is written into a contract that all injuries to animals and humans are the dog owners responsibility - and even then the contract should specify that includes animals owned by the sitter. It is unlikely that such clauses are in 99% of contracts and no-one should sign one that says anything like that. If you have no contract as I know is the case with a lot of sitters - she has not a leg to stand on.