hm.. All I will say, is definitely wait to take her to the park or basically anywhere with distractions for now. She needs to have solid recall, sits, downs, stays, heeling, and everything in a familiar environment before you move on to training her in public places. Start by training her inside of her house or in the yard, using a LOT of tiny hot dog bits and praise.

It must be really annoying that they're asking you to train their "out-of-control" dog, and they wont even buy a new gate for her -.- ... that would annoy me, at least. Tell them they need to secure their backyard if they want to improve their dogs behavior. She sounds like an under-stimulated and under-exercised puppy. Puppies her age and breed usually need 1-3 hours of good exercise every day! That's probably a big part of why she's bolting out the door and out of the yard. Try playing ball, tug, frisbee, or run around with her for 1 hour before training. Let her rest for a while, then start training. You should have your Aunt sit outside and watch you train her, at least.

Also... halti's could possibly help with walking for now. You can order them, chokes, and prong collars on-line. A fine choke chain that fits her (it should fit her neck with only a few inches of extra chain) could help with training. Adjust it so its always around the top of her neck, where it meets her head. Don't pull on the choke constantly! Only pop it once to correct her when she pulls or ignores you. Martingales also work really well. Good luck!! It sounds like you have a really tough job ahead of you... it'd help a lot if your Aunt called a behaviorist for professional advice or you looked for better Obedience classes.