Not sure whether this will help...
My youngest learnt to cycle after one time when he was tired I asked the next child up to let me push him on her bike for a bit. I noticed he was able to pedal and balance. However it took 3 months from that moment to when he realised it himself. Those 3 months were spent with me walking/running along beside him pretending to hold him, him checking I was still there. Anytime he realised that I had let go he would slam on the brakes. When he started school we did the same routine to school every day, until I finally convinced him that he could do it - and that was the start of him cycling to school daily.
The next child up from him was very different, she only ever pedalled half circles, then back pedal half circle etc, whilst on stablisers. Eventually in desparation age 6 I took the stabilizers off at the beginning of the summer holidays. I did a deal with her that we would try each day for 5 mins, and after 4 weeks she was still doing the half pedal thing unless I held her foot and made the pedal go in a circle. It was a stressful time as she is my most awkward child... but luckily 5 mins is short. Then one day she got it and could cycle about 5 metres - it was a real breakthrough moment. For her especially I wish I had known about the removing pedals at the time.
So I'm not sure what the above is meant to tell you... keep perservering?
A friend of mine used to run sessions at her local school to get kids off stabilizers as part of Safe Routes to School, and I once went along and watched their sessions. She promoted it as a fun session (drinks, biscuits etc), just give lots of reassurance, don't mention falling as that is the only word they will hear, and you were meant to ask their permission before you let go. (This was after I had taught all mine to ride).
I'm sure there are trainers out there that would teach a 6 year old, but before trying that, is there someone else in the family or friend that would maybe give it a go teaching her, to remove the personal element. The daughter mentioned above had her bikeability training this year - I know I could of done it, but equally knew that she would behave better for someone else. As it was she spent the entire time talking about only doing it for me and that she hated it... but I called her bluff and said she didn't have to do it if she didn't want to (not what she was expecting at all). She's like a kitten who has to keep prodding the sleeping lion trying to get a reaction.