The best thing I ever learned was how to listen to my instructor when poo hit the fan. At first my self preservation instincts would take hold and I would find myself either on the ground or in the saddle by chance. One time poo hit the fan and my instructor belted into the microphone "NOW LISTEN TO ME!" It was loud enough to open my ears outside of what was happening on the horse's back she she said it with such authority that I suddenly became more afraid of NOT listening to her than I was afraid of what was happening on the horse.
A little fear and respect of your instructor will go a long way!
Once I learned to listen to my instructor, I realized that I was saving my own butt. I automatically searched for her words when poo hit the fan and she would work us through it. It eventually happened enough that it became second nature, I was able to bring the horse back without her saying much at all.
I do remember this one time where a school horse was bucking a student off and I just so happen to be in the arena for jump crew. She told me to get on and I used the student's helmet, which was way too big. But hey, it was just a couple bucks and I didn't feel like I really needed a helmet, so I put it on. The horse LAUNCHED into the air with a big leap and when we landed, the helmet fell over my eyes. No time to pull it back because he was broncking around. I had no sight and wsa about to fall off to the right. That trust I had in my teacher and my listening skills saved me. My instructor belted into the microphone and my ears picked up "Left seatbone! Sit back, sit back, pick his head up, low right seatbone, you're leaning too far in!"
The entire ordeal lasted a couple seconds but she talked me through a bucking fit while I was literally BLIND. I'll never forget that ride. Once the fit was over, I switched the helmet out for one that fit better and continued and the horse fought a bit more and then settled in.
As an instructor, you need to talk your student through what is happening and the best thing you can tell your student is "Whatever the horse is doing, just ride it. The horse is running away from you at the canter? Then you're cantering. Just ride the canter" People tend to panic when they aren't in control and the gait wasn't their idea. Teach your student to just ride the gait. Its just another gait.
And then you have to worry about your student listening to the advice
I know you feel bad, but you probably did everything you could to work the student through what happened. Now that she knows what CAN happen on a loose rein contact with this horse, she just MIGHT pick up the slack! Bad experiences have a tendancy to make us take the advice we should have taken before the experience!