I agree with pretty much all the previous posters - get your daughter on a lunge line for future canter attempts, AND give her some space about the cantering issue.
The other thing - I'm not sure why she's riding a pony that is, though bombproof, prone to keeping on running when she "bails" mentally? Not all horses are able to be a scared kid's horse and it might do to find one that can handle her breakdowns [and STOP when she stops riding].
I teach lessons on my mare and that's perhaps her best quality as a lesson horse - doesn't matter what happens, if there's a child on her back and things go sideways, she WILL stop rather than bolt. With me, an adult, she'll totally bolt if the mood strikes, but absolutely never with a child. The kids know this and it gives them a LOT of confidence in her because they know that, even though the new activity might be scary/intimidating, they only need to worry about themselves.
The kids always start working a new gait on the lungeline and I have no qualms about throwing them back on the line if they need a little help. :) It gives them less to focus on and I think it helps my horse not get frustrated with beginner mistakes.
My mare actually worked her "magic" on me too - I used to be TERRIFIED to canter due to a series of bad experiences when I was taking lessons [I rode a gelding who would, without fail, buck into the canter. He got me off probably every other lesson and it really scarred my cantering experience] but then I started riding my mare, I realized that the last thing she would EVER do was buck at the canter or try to intimidate me at any gait higher than a walk.
Now I love cantering, we canter a lot. :)
On other horses, I'm still nervous about cantering but I don't directly go to panic-land like I did when I was younger.
Not pushing the canter thing and letting me take my time getting there worked for me. However, with your daughter, I think I would reallllly try getting her on the lunge line and, if possible, getting her on a horse/pony that'll stop if she freaks out. That honestly has to be pretty scary - to panic, lose control due to panic, AND have the pony keep going... I would avoid cantering too, honestly.
She'll get there one day!!