You said that you paid her the money back?
Who is paying for the new car?
Did you tell her you are not paying anything for the car?
You didn't do anything wrong. I understand that she lent you the money, but that doesn't mean that she can control your life. If the older car wasn't working, then you had to make a decision. Although she did not like your decision, she should still respect the decision you made, as you are an adult and it was yours to make.
If you've already spoken to her and tried to explain things to her, there's really not much else you can do. You can start speaking to her and 'killing her with kindness', meaning every time you see her, smile and act cordial. Give her compliments here and there, pick her up some coffee on your way in, and just be really nice to her like nothing ever happened. Not sucking up, but being nice as if you were still friends. Asking how she is, giving her a sincere compliment, even if she still acts standoffish, just ignore it.
After a while, she will see that her meanness and anger is not hurting anyone but herself. Even if she does go back to acting normal, I would be careful about letting yourself get too close to her again and I certainly wouldn't borrow money from her again, since if you do, she wants to control what you do with it.
I welcome your thoughts, let me know if you want to talk more