How would you deal with an immature/over-selfish friend?

Just before you read: if you don't want to read the detailed below, I basically just want a little advice what I should do about my supposedly best friend who's being overly immature and self centred, to a degree that I'm being very negatively affected by it; my sick mum wants to be sure I have her as a resource, and she's also been my best friend since I was 14.

To those who want to read the details, remember, it'll feel like I'm whinging.

Basically, my best friend started off being very mature when she was in highschool, but lately she's been very "ME ME ME" in a way. Mostly it's been just, that she'll want me to hang out with her when I've already made plans with my partner (which has been hard, considering he's been overseas for a few months). She also picks fights with me whenever I do something she doesn't like, and that's whenever I do something that isn't her way.

I guess that's not the issue: she admits she's selfish, and I'm selfish at times, as everyone is; but it's got to the point where it's affecting me badly, such as...

When my mother was in hospital and I was distraught (due to why she was in there and how far away she was), she promised to come and see me to get my mind off it. The day before, she said she was going to a party that night and wouldn't be able to see me until the night after, and only for about an hour, because she wanted to stay at that friend's place.

When I confronted her, instead of hearing "She was in a really bad way and needed me there, I didn't think you were that bad off, sorry" or something like that, I got "Well, you never hang out with me for that long because you talk to your partner everyday, so I'm teaching you a lesson by hanging out with her!" I do talk to my partner everyday because he's oversees for the next few months with family troubles; even then, I've been late to talk to him because I've been hanging out with her all morning, and have left early to hang out with her for the night. I've even not talked to him when my best friend needed me.

That's just one example. She's done more since then.

What should I do?

Thanks for any advice!