Some budgies do have "hand issues" and simply don't want your hand anywhere near them.
Yours may well be one of those.
If you do decide to start the taming/bonding process from the beginning and on a regular basis, then continue trying to hold your hand outside the cage where she can see it but not get to it. Perhaps when she is interacting with your face, hold it near your face?
Yes, if you progress to the point where you are able to put your hand in the cage and she begins biting, then you go back to outside the cage again.
I'd definitely make a bird-safe room and try giving her the option of coming out on her own.
You may well have much better luck in interacting with her outside the cage rather than in it.
I have untamed budgies that go back in their cage because I've used positive reinforcement training to teach them they will get a reward when they do so.
Initially, when I was ready for them to go back in the cage I would first dim the lights and pull down the shades in the room. If the radio or TV was on, I turned it off.
Then I would stand by their cage and ringing the bell on one of their toys while telling them, "It's time to go in your cage now!"
Once they went back into the cage, they were given a small bit of millet.
This method worked well and they now go in the cage when asked to do so without the lights being dimmed or the blinds lowered and the TV or radio can stay on.
Getting the millet reward is the best part of "going back home"