Balance and rhythm exercises?

My ottb mare has some balance and rhythm issues because she's been out of work all winter, minus some hacking. So far each school has been 30 or so minutes. I've been keeping it short and simple and want to start from scratch.

Now, our biggest issue is rhythm. She's fine at the walk, typical strong thoroughbred walk, but as soon as we get into the trot her rhythm is all over the place. It varies from short/choppy/lazy strides, prancing around, trotting sideways, or taking biiiig sloppy strides. Her canter is more consistent, but occasionally she'll brace and will rush out of nowhere (balance issue). I've only been taking her a few strides down the long on her right lead because it needs a ton of work and she's super unbalanced in this direction.

SO my question is, what are some good exercises I can use to help with her rhythm and balance issues? I'm in no rush, so any advice is greatly appreciated.

Oh and a little photo before our ride today

Pretty girl. Does she have the same balance issues trotting without a rider? I would say keep her on a large circle to help her find her balance, first without a rider and then with one. Once she starts to find her basic balance, add a ground pole for her to step over each time she goes around, and then work up to trotting series of ground poles and raised cavellities.

I would lunge her on a line mostly at the trot with side reins (Not to tight)

When riding her I would work on a long rein and again mostly at the trot. If she rushes and starts to canter, I would guide her onto a circle, sit heavy in the saddle to make it harder work for her and when she comes back to the trot, start rising and work large.

If she is lazy then I would push her forward.
Racehorses are use to a constant pull on the reins and when they do not get it they will panic and go faster. Use your seat and body to balance her, you can regulate the trot by rising slower and lower to slow down and higher to lengthen the stride.

Thank you for the input!

I've been working her on a long rein so she can learn to support herself rather than relying on my hands. Also been doing some hill work and downward transitions to work on those hind end muscles and getting her off the forehand and have seen a wee bit of improvement but it's only been a few schools.

Lunging actually totally crossed my mind, I'll try it without the side reins on Monday then introduce them another day.