Cheri, I agree with the others in regards to not forcing her out. Forcing her out and then forcing your attention on her is a form of flooding. Does it work? Ya, of course it can! But it's hard to say if you end up with a bird that is truly happy or a bird suffering from something akin to stockholm syndrome. It's all about a state of mind, and unfortunately, we're not mind readers, and humans, in general, are pretty bad about understanding animals, let alone parrots!
When another member mentioned sitting as close to her cage as possible that she's comfortable with, it doesn't mean sitting next to her cage, but it could mean sitting 3' away or maybe 5' away. Maybe even further. One exercise you could try is walking towards the cage slowly and the moment you notice that she's looking scared or terrified in the slightest amount, stop. Let her relax and calm down. Then, take a step back. Repeat the behavior of walking towards her and stopping before she is really stressed out. This includes shaking her feathers, a behavior we want to try and avoid. Basically, trying to get you to appear as none-threatening as possible and letting the bird tell you what she is most comfortable with. With this exercise, it may mean that you can't go over to her cage to open up the door. You might need to use this same exercise though for changing her food and water bowls. If she appears stressed in the least, back off a moment before proceeding.
Hillary Hankey has a couple of great articles on trust which goes a bit more into detail.
Working with Fearful Parrots: A Study in Videos | Learning Parrots
Parrot Behavior Myths: Building Trust | Learning Parrots
kozykitty, If you are going to open your zon's cage, I would highly recommend putting your cats up in a room where everyone will remain safe.
What I would recommend for her is two things. One, station train her. Two, target train her. If you station train her to stay on a specific perch away from her dishes, then you can change her dishes without getting bitten. At first, you may need to lure her to a specific perch and then start feeding her her favorite food over and over again. Make sure it's small pieces though so she eats them quickly and is wanting more! Once she gets the idea that staying there equals good things, then you can start messing with her dishes (not taking them out, but touching them, turning them, moving, etc) with the other hand and continue delivering rewards with the other. Over time, you can start feeding her less often, slowly building up a "stay". Instead of offering a treat every 3 seconds, make it every 5 seconds, then every 8 seconds, then 12 seconds and so forth. You'll be able to get to a point where you can tell her to station, reward her, remove her dishes, reward her again, clean her dishes, refill them, tell her to station, reward, replace the food dishes and reward yet again.
Target training is kind of similar, except we want her to orientate her beak to a specific item. It could be an acrylic stick, a knitting needle, an empty pen case, or any other object! It's a little easier if you can find something she's not afraid of, but if she's afraid of something, you can work with that, too! Just might take a little more time.... the main idea here is to have her look at the target, reward. Move her head towards the target, reward. Stretch her neck towards the target, reward. Take a step towards the target, reward. Take another step towards the target, reward. Once she gets the idea, you want her targeting around the entire cage and if she does that reliably, then you can open up the door and start target training through the door! If she gets that down pat, then teach her to come to the door to target (may help to have a perch attached to the cage door, and even some perches on the outside of the cage!), then eventually target training her around the outside of her cage.
Step up is actually a form of targeting, so you could target her to walk closer and closer to your hand for a reward, and eventually even touching your hand! And that could lead to teaching her to step up!
If you start training from inside the cage, you may be able to avoid cage aggression! Which happens a lot to birds who are trained away from their cage as it usually involves forcefully removing the bird from their cage then they go and train. This only teaches the bird that leaving the cage can be a traumatic experience and they're going to bite to tell you they don't want to leave on *your* terms!
, and that is a difficult issue to overcome. Cage bound birds should NEVER be forced to come out, just provided the opportunity (with perhaps a little enticing) to come out ONLY if/when they WANT TO. I would suggest doing just what you are, opening the door every day while your there and letting her take all the time she needs to work up the courage to come out. This could take weeks or even months. If she has a favorite treat or toy, you can always put that right outside the cage so she has some encouragement to come out. 
), is to sit as close to the cage as she seems comfortable with and talk to her in a calm, soothing voice. You could even read a book out loud to her. Doesn't really matter what you say (she doesn't understand words), it's all about the tone of voice. If she approaches you, offer her a treat and a (calm) "good girl!". You have a lot of trust to build with this one, as it sounds like her prior owner(s) never gave her a reason to trust humans. Best of luck, and remember be patient, persistent and let her come around on her own time.
