Ouch! That is a bad ride.
A small round pen is fine as long as your horse can canter around without much risk of slipping and hurting themselves, but she's pretty small so I hope it's ok. Be careful though! I've seen too many almost accidents even in decently- sized round pens.
Round penning advice:
I use Clinton's method of round penning - I actually learned it watching his videos and clinics. First, I establish direction and speed. I point in the direction that I want the horse to go, and most likely the horse won't get it the first time but eventually just responds from a point.
Then, I make sure I have my 4 ft stick with 6 ft string attached so that I can create energy while staying at a distance - which sounds important with a horse like yours. I increase the energy towards the pointed hand until the horse goes - then I release the energy at once. It's important to keep increasing energy until the horse realizes it has to move in the direction you want her to go. It's easy to be intimidated if your horse starts to freak out, but remember that you are communicating something to her, as it's likely she has never done this before. If you stop once she starts to freak out and do the wrong thing, that's what you are teaching her to do. Instead, teach her to move off of the pressure by only releasing at the right time. Be safe, though! If she starts to come towards you, focus your energy on keeping her out of your space. I always start this exercise with a new horse while wearing a safety vest and helmet, even on the ground.
Also remember to apply a tiny bit of pressure and work your way up, Don't release it at all until you get what you want, but start small so that eventually your horse will respond simply from a point. This will give her a chance to respond. Clinton has a great "rhythm" method - start with 4 rhythmic strokes tapping the air, then 4 rhythmic strokes flicking the air with the string, the 4 strokes smacking the ground, then 4 strokes REALLY smacking the ground while stepping towards your horse, etc...
In round penning, when you are applying the pressure, you must stay behind the horse's "drive line," which is the place where the girth goes, right before the shoulder. If you move in front of this the horse will think you are cutting them off, as the pressure shifts, and will try to turn around. A lot of my students do this without realizing it and then try to punish the horse for turning around. They're just doing the right thing, and it's easy if you're not used to round penning to let yourself get in front of the drive line in a small circular space.
I correct a horse, however, when they try to turn around when I don't ask them to. I also correct them if they do something rude such as kick out or pin their ears. I do this by flicking the whip, pointing, and asking them to go faster. Usually I say a vocal command like "NO" in a big voice so that they understand - after all, I am simply communicating to them what is right and wrong. My philosophy is that threatens my safety is worth correcting when working with a 1200 pound animal as long as I retreat when they change their attitude.
Once your horse has clearly understood that they need to be moving forward quietly and respectfully to one side, it's time to control their direction. Backing away should get her to come in towards you, then you shift to the other direction, switch your whip, and ask her to go. However, this is ideal, and usually only a very well trained horse will do this. If your horse is smart she will pick this up quickly but it might take some patience. Sometimes your horse just does not stop, or faces their hind quarters, not head, to you, in which case you can just cut them off to switch directions. Once you've got the horse changing directions and speeding up and slowing down at your command, congratulations, you are the "old mare" and leader of the herd!
Then, stopping and asking the horse to join up is a whole other art: the idea is to back off. Ideally the horse will stop and face you, and then walk towards you. You let them sniff the top of your hand (hold it out to them so that it's shaped like a muzzle) and stroke their necks (this is how I learned to do it, apparently the mare does this in the wild), and walk off. The horse should walk off with you. He will be joined up.
However, in practice, you don't always get this. The horse will face his hindquarters towards you sometimes, or not come to you. Here is what I (and I think Clinton) do: I make myself as un-predatorlike as possible by turning my shoulders away and looking down. I face their shoulder instead of walking directly up to them or behind them. I step forward 2 steps, and then back off 3 steps and call them to me. Most will look at me or take a couple of steps toward me at that point. That is a try, so I reward that and come to the horse. If the horse looks away or further faces their hind end to me, I send them off and round pen them again for a couple of minutes before I try again.
Once I have arrived at the horse's neck, I expect them to follow me, however slowly. If they don't follow me at all or randomly leave, I send them off again. So watch out how much you ask for the first time, when they are still learning - only one or two steps is usually enough. I usually just walk back to the lead rope, and then clip it on them again to start leading them regularly! Eventually ask for more and more as the begin to understand.
Don't give up until you get the pleasant reward you want unless your horse appears hot or out of breath. Remember, it's likely she's never done this before and you are communicating something to her.
I hope that this novel helps. I made this public so that people can share their experiences and opinions or maybe benefit from this. Does anyone do this differently?
However, you can private message me with any questions. Also, I'd love to know how she goes!
Let me know if you want any ground work videos! Sometimes you don't understand until you see it. I don't think we have a round penning one yet (but You Tube does!) but we have plenty of other respect ones.