How do I say sorry to my ex girlfriend after I broke up with her?

Me & her was going out for about 4 months but we knew each other for much longer than that. She had a big crush on me abd I was single at the time. We would talk on the phone alot, I'd visit her, she'd share alot of deep secrets with me, telling me about having a rough childhood and everything. It was like she needed a friend but also she was falling in love with me very fast. At the same time she had a boyfriend, but whenever he did something wrong, she'd use that as an excuse to dump him just to be with me. Well anyways, me and her started to date. Things were fine at first. It wasn't perfect, we argued alot, she'd take things I say & twist them around or I'd say something to hurt her because of how immature she would act. Well I had broke up with her a couple of days before our one month anniversary. She cried badly and begged for me back. I felt horrible so I apologized and took her back. Things started getting better between us after that. Then about a month later she broke it off with me saying that she didn't wanna be in a relationship anymore. I was heartbroken, I kept asking her why she did it and she didn't have an answer. So later that night she txt me and asked for me back saying that she was stupid for breaking up and that she was still in love with me. I believed her and went back out with her. Now the month after that, I didn't have alot of time on my hands. I had alot of family problems going on and I havent seen her in over a month but we did talk all the time over the phone. She also tell me that her and her ex boyfriend who did cheat on her became cool again. He lives downstairs from her apartment to and she still hangs out at his house sometimes. Christmas Eve, she writes me on AIM saying that she doesn't feel the same way about me anymore. I take it as a break up so I sign off. I don't hear from her until New Years day. I asked her what was her reason for breaking up with me and she says she never broke up with me. She says that I took what she said the wrong way and that she should have called me to get me to understand what was really going on. She was upset at the fact that I don't visit her enough. So we talked and things went back to normal, so I thought. I told her I couldn't see her for a while and after that, whenever I'd called her she'd say she'll call me back but didn't. I'd write her and she'd give me those "one word" answers. It's been going on for about a week. I would ask her why she never returned anything of mines and when she did respond she'd say that she was too busy (she stays at home on Twitter & AIM all day so I know she's not doing anything). I called her and she hung up fast saying she'll call me back. I wrote her on AIM and told her that I was dropping everything I was doing to come visit her and she said not to come over. So with that added on to how she's been playing a distant role decided to break up with her. It seemed as if she took it very hard. I felt bad about how I did it and tried to get back with her but this time it wouldn't work. She said she wanted nothing to do with me anymore, then blocked me over AIM. The next day she unblocked me and I figured she was giving me a chance to say whatever I had to say. I wrote her and she cursed me out. She said how much she hated me and that I was the worst person on the planet then she reblocked me. I felt like how can she tell me that after and all I wanted to do was apologize, so I txt her and said some things that I should have never said. She didn't respond. So later that night her s/n popped up on AIM. I wrote her the next morning asking her what was her reason for unblocking me and kind of dissed her. I told her she should leave me alone and she wrote back and told me that she would for good. Everything between me and her was fine before we started going out, I asked her if she still wanted to be friends and she said no to that too. She says that she wouldn't block me over AIM, she'd just delete me and all of my numbers and pictures. I hurt her badly I know but it wasn't as bad as her ex, well maybe she made whatever her ex did sound worst than what it was. I want to say sorry so bad and I feel terrible because of the way she reacted. I know how she is and she's been disrespected before by ex's and famil. She put what I did to her over all of that. I just want to know how do I say sorry or should I give her all of the space she needs to think and recover?