There is no bad horse, just bad owners. If your horse is acting this way, it's because of a previous owner. It seems to me that if she is acting this way, she is either doing it to protect herself or she has just been allowed to and nobody has corrected her.
Your mule, who you say, "now has her manners" has them because you have not corrected your mare and has picked up on this. That being said, the same will eventually happen with your foal if you don’t show authority. If you plan on keeping the foal, ground manners/training should be taught to your foal as early on as possible- the sooner the better.
About your mare, if you do wish to keep her, you need to have the time to. Horses are a huge time commitment and if you keep either mare or foal, you need to have time (and lots of it).
That being said, you need to start laying the law down with your mare NOW. The longer you let her keep acting like this, the worse she is going to be. This doesn't mean acting all macho and tough, you just need to stand your ground and correct her when she lashes out. If you are going to keep your mare, I suggest consulting with a trainer and a veterinarian if you do keep her, just to be on the safe side.
I wish you the best of luck :)