I had a very similar thing about the same age-I wasn't bothered by needles in the slightest until I fainted after a vaccine, and after that, I just went crazy around them, to the point where I once fainted when my medical student cousin just told me a story involving needles! This went on for a couple of years, until somebody suggested trying the following, which hopefully might help your daughter.
Get your DD to lie down on her bed, or on a couch or somewhere else she feels safe, and to imagine she's about to have an injection. Then to imagine the needle going in, and then coming out-chances are she'll feel faint and sick, but she can rationalise the fact that even though it makes her feel funny, she isn't going to come to any physical harm. The thing that helped me was that it was a horrible image, I felt weak and dizzy, but in reality I was fine-it wasn't doing me any damage ultimately, IYSWIM? The lying down means she's far less likely to faint, and if she does, she's safe. She could do a similar thing with her orthodontic treatment, too. A glass of water next to her can be helpful too.
Over time, this improved the phobia massively, and I then started making myself watch YouTube videos of injections and blood draws, and again rationalising that whilst it was a slightly painful procedure, in the grand scheme of things I was fine. It took a while, but it improved massively over time and I now work in a hospital, on the other end of the needles, with no problems whatsoever! As for your 6-year old, when we get younger children in the hospital who are a bit nervous, we let them "inject" one of the staff with a syringe tube without the needle in-could you let your DC practice with a toy doctor's kit or similar?