With a Jack, you're going to have to have an intense excercise schedule too to keep her from have energy to do mischief. I really recommend walking at least an hour with her and then rollerblading/jogging/biking maybe 15-30 mins ontop of that to really let her run her energy out. Training anything is MUCH easier when they don't have energy.
What I would do in your situation would be to get her tired, and then clip on the leash. Then, get her and the cats into a room but hold onto your dog. And then with treats make her do tricks and such infront of the cats so she gets used to not paying them any attention. In fact, good things happen when she doesn't pay them attention. And repeat this several times a day. But I would talk to a professional trainer first though, because there may be a better/more efficient way to do it.
I disagree about not correcting behaviour. My dog gets all the possitive reinforcement in the world, but when she does something really bad and unacceptable, she gets "grounded" aka pinned to the ground gently (it doesn't hurt the dog at all, but forces them to relax). She tried to bite me once, I grounded her and she's never done it again. Likewise with trying to nip at children or lunging at a dog for no reason. I'm all for possitive training, but I think there are things that call for a correction, just like with kids. She's now a fantastic adult dog, super well behaved (and definitely not traumatized by her corrections.) If she ever bit/hurt another pet I guarantee you she'd be grounded in two seconds. Also talk to a trainer about learning how to do it, because there's a trick to it.