I can be her best friend, or not her friend at all?

(This is a long read, but it is a bad problem I'm having, so advice is appreciated as I also want to know whether I did the right thing :)

I've had a friend in secondary school for about a year and a half (we'll call her Katie). I just started this school last year to do sixth form (the last two years of school), and I began to bond with this girl because we both have health problems and I have no other friends with problems like mine. We became very good friends very fast, but I discovered from her visiting carer (who just checks up on her now and then) that I'm her ONLY friend, not just that, but the only close friend she's EVER had. I was flattered at first, but after six months I discovered the effects that having little to no friends has had on her; she would spit at me and find it funny, and make fun of serious issues like terrorism or anything to do with the racial diversity in my school, things that were so immature and shocking, that I had to ask her to repeat it to make sure I'd heard right.

She would also cling to me and follow me everywhere, or get jealous if I talked to other friends of mine. When we were standing talking and there was a gap between us, she would step into it and be in my face, as if a few inches of difference disturbed her.If I stepped away she would do the same, again and again and again, but would never get the hint.
I'm not a very healthy girl, and I'm being diagnosed for a problem, so I'm in hospital A LOT, not sleep visits, just appointments. This girl has my number, but I told her I only use my phone for emergencies, and I mean that. I told her straight up that if she would text me with something she could tell me at school, I would ignore her, because paying credit on my phone is expensive. And I would tell her when I was going to the doctors or going to be off for a while, and she shouldn't call me because I would most likely be fine, just resting. She agreed.
Then she would call or text me EVERY day with claims of being worried, and that I'd not made her aware! And I held firm to my promise not to reply, but then she bombarded my father with calls if I was at his house, or my mother if I was at hers. I did talk to her the first time she called my father's house, and told her not to call as I was tired or not well, but she would continue, and it made me more angry than it should have. Because I was going through a lot of tests, and I needed space to recover, because it was rough stuff I was having done and I just did not feel like talking.

When I got back to school, she said she wouldn't she leave my side.

Whether I liked it or not.

At this point I was so angry and tired, that I told her, as gently as I could, to leave me be for a while while I recovered. But she didn't, so I had to ask her mother to tell her not just about my recovery but about everything else she'd done. I was too impatient to do it and she's the only one who could explain the situation to her. I couldn't cope with her clingy, possessive attitude towards me, it was stifling me and making me angry and annoyed with her, and I knew I would be too short tempered and harsh with her.

Her mother talked to her, and she refused to talk to me ever again, seriously, she will not say a word to me now. She even refused to welcome me back after summer when I tried to be friendly with her.

I was just wondering, should I have approached her differently? I've never had such friends, I've always been more introverted and I couldn't cope with the stifling attention. Keep in mind, I wanted her to be my friend after her mother spoke to her, but she behaves like it's best friend or no friend at all. Also, is it normal and eighteen year old girl behave like this?