Ezra, your comment has clued me in a bit. I think we have a bit of a self-confidence issue here...for your mare. Her safety zone when you are handling her is next to you. And she is more than a bit defensive about being forced to leave that safety zone. This is actually a pretty common problem, so we are going to fix it before worrying about other stuff.
Actually, last summer, I had the very same issue with Turk, the paint gelding. He handled fine, but when I started working with him and playing some of the Parelli Games, he freaked on the left side. Either he was trying to run over the top of me or he was trying to leave. It was no fun for a few days-he is a hard-headed bugger.
(In spite of what some may think my take on the Natural Horsemanship movement is, I thought Pat Parelli's original 7 games was very innovative when it came out and I use the methods regularly.)
I like a rope halter, a long lead rope and a short buggy whip. Usually I start working on the good side first, just to get the feel of everything. Take your mare to the center of the arena and move to her right shoulder. Put the leadrope in your right hand, the buggy whip in your left hand and step straight back from your mare. With your body facing your mare's shoulder, move your right hand out to the side and cluck. She should move forward, following the line of the leadrope. If she tried to move her shoulder toward you, poke it with the buggy whip. If she goes so far as to swing her hips away from you and tries to come toward you, you will have to move your feet to follow the shoulder-keep your hand out, showing her the direction you want her to go. Keep clucking to tell her you want movement.
There are two things that horses usually do right about now, if they haven't willingly followed the leadrope and maintained their body distance from you...
1) They may rush past you. That is okay, if you are worried that they may kick out as they rush past, simply drop your hand and pull them around to face you. At least you got forward movement, which is your initial goal. Or they will...
2) Start spinning their hindquarters away from you. This is a bit harder to keep up with, because you will have to keep moving your feet to keep your position even with their shoulder, while maintaining the right arm out to give them a direction to head and tapping them on the hindquarters with the buggy whip. And clucking. We want those front feet to move. Anytime they move the front feet AWAY from you. Stop. Regroup and ask them to move again.
Pretty soon you will have a horse that will walk a circle around you, maintaining the distance you desire. Then you can move up to asking them to trot around you. You should be able to bring them closer to you by shortening the leadrope or push them away by lengthening it. Gradually you can work the circle out to the end of the leadrope.
Now for the bad side...
This mare is already exhibiting dangerous behavior. She has managed to figure out that there is little a person can do to control her movements at the end of a longe line. So the goal is to get control of her movement and make her keep those front feet moving in the direction you want them to move.
Again, take the mare to the middle of the arena. Place the leadrope in the left hand and the buggy whip in the right. Stand at your mare's shoulder, facing it and step back. Lift the left hand to give her the direction you want her to head and cluck. Most likely, she is going to want to turn toward you, keep moving your feet to maintain your position at her shoulder, make sure to keep your left hand out and keep clucking. Poke her in the shoulder with the whip if you have too. The biggest goal is to get her to move those front feet, AWAY from you. At first it might feel like you are the one getting the workout. If at any point you feel like you need to stop and regroup, by all means do so. Maybe lead your mare a step or two, so she moves those front feet. And then ask again.
If she gets excited and starts to back up, go with her, try to maintain your position at her shoulder, but if you can't, just keep backing her until she is controllable again and stops because you ask her too, move immediately back to the shoulder and ask her to move forward again. The secret is to turn the backing into your decision, not hers. If she starts flying backwards, go with her, keep her backing until you feel in control again, back her some more, step to her shoulder and ask her to move forward. As long as she isn't backing into something that could injure her, don't worry if she backs into walls or backs the length of the arena. Pretty soon she is going to get tired of backing up and will try to find another way to get away from the pressure you are applying-hopefully that will be in the direction you want her to go-forward.
Now for the rearing part of it...I'm going to tell you how I handle it, but if you are not comfortable doing that...DON'T DO IT. Your safety is of the utmost importance!!! For a horse that gets angry and starts popping up in the front end or even rears all the way up, I let out some leadrope, step back and off to one side and yank the horse's head toward me. As soon as they get their front feet back on the ground, I make them start backing, as fast as I can make them go! A horse that has all four feet moving has a hard time rearing up.
I have never subscribed to the theory of pulling a rearing horse all the way over. I have seen people do it and there is just too much room for damage to both horse and handler. Besides it doesn't really seem to work. It doesn't teach a horse how to avoid rearing by making them move their feet, it just seems to make them a little more unpredicable when they do rear up again.
The alternative to pulling your mare toward you when she rears up is to just let her and as soon as she gets those front feet on the ground again, get after her to move her feet, either by backing or again trying to get her to move forward.
The turning the butt to you and kicking out...well, that is a horse's natural defense. On a leadrope and in closer quarters it is easier to control. When she gets mad and wants to tip that hindquarter toward you, the goal stops being about moving forward in the direction you want her to and becomes making her move that hip away from you. Bring your leadrope hand back to a normal position and start popping her on the hip with that buggy whip. Use the leadrope to keep her head tipped toward you and keep whacking her in the butt until she moves that hip away. Make it sting too. In this instance, you have to be the lead mare and no lead mare is ever going to tolerate another mare's aggression. Don't stop until she moves that hip away from you. When she does, regroup and ask for forward movement again. Eventually, little miss thang is going to get tired of doing anything but what you want her to do. You are going to turn all of her antics into hard work, not escape routes. She is going to have to start thinking about what you are asking and actually start processing the cues you are giving her.
I'm guessing you know full well how the standing at the shoulder position works, since you do longe your horses. But the nuances are a bit exaggerated when working in these smaller circles. Standing square facing your horse's shoulder is a neutral position. You aren't asking your horse to move. When you step back and lift the hand holding the leadrope out, you are opening up the door as the logical direction for your horse to move. Your leg on the same side that you are holding the leadrope should turn in the same direction you want your horse to go, so that your toe is pointing that direction(as if you were going to step that way). This makes the other side of your body turn toward the horse just a bit. If you were to raise the other arm straight out in front of you, you would be pointing at about the girth area. Effectively, with very little body movement you have pointed out a direction for your horse to go with one hand and applied pressure to his body, implying you want him to move forward. The buggy whip is just an extension of the pressure you are creating with your body position. It can be used to poke the horse in the shoulder or girth area, encouraging him to keep his distance or you can open that arm out and tap him on the hip to encourage forward movement.
She may do fine in close quarters and only start having trouble as she gets toward the end of the leadrope. Just keep building the distance and reinforcing that nasty behavior causes a lot of work for her. As it gets easier to send her out to the end of the leadrope and she maintains forward motion for you then it will be time to reintroduce the longe line and build to even more distance. I highly recommend looking up Parelli's Original 7 Games for other things you can do to build your mare's self-confidence too.