I will add my encouragement to that of other posters. As jenluckenbach said, there are no guarantees of long life with any pet, no matter how heartily healthy they are when you adopt them. When I was talking to the emergency vet last year about my 17 year old Artful Dodger, in tears because he was dying "so young", he said that he had to help people cope with losing their 6 or 7 year old cats to cancer every day.

As far as I know, and others can correct me here, an FIV+ cat, unlike an FLV+ cat, does not run the risk of getting cancer more easily. As others have said, you do need to be aware that if this cat appears unwell, you shouldn't wait a day or two to see if she get better by herself (as sometimes happens) but take her in right away. My Sesat had feline leukemia but lived for 7 very healthy and happy years from the simple recipe of copious amounts of love, no stress, and fast visits to the vet. FIV is far, far less debilitating and I wish that she had had that instead.

Can you offer to foster this cat to see if she and your dogs will get along? Or am I suggesting something too stressful for both you and the cat? I really hope you decide to scoop her up and take her home.