So frustrating, isn't it? At this age, they start to assert their independence (ultimately a good thing), but they're still scared about not being protected by Mom and Dad, and they're old enough to identify your weakness (like being hurt by the "I Hate You" song).
You don't have to go to every fight you're invited to. I think you can do reminders about things, but really, what horrible consequence will result if they don't get to something on time or they aren't dressed appropriately? What horrible thing will happen if they don't get their homework done one night? Let her go to things and face the consequences, either with the teacher or the activity advisor. If she had a key role in a school play, then not being there would have a terrible impact on the others, but if it was a chorus concert or other group performance, so what if she missed it? The world wouldn't end. She'd either be disappointed and wouldn't do it again, or she wouldn't care and you'd know not to sign her up for something in the future.
When my kid didn't want to get ready for school, I nudged a lot. Then it occurred to me, So what if he's late? He's in 3rd grade. I said (forcing myself to be calm), "Okay, that's fine. You're too big for me to carry. But just a reminder, if you arrive late, the school rule is that you have to check in with the office. So when you're there and they ask you why you are late, you can give your reasons directly to the principal yourself." Brief look of horror. Quickly got dressed.
He did this occasionally as a teen. He mouthed off and was, I thought, disrespectful of adult rules. So I told him I assumed he was too immature to drive. I took the car keys and license plate off his car (my mother had given him her old one), and told him what time the school bus arrived. He was defiant because he thought I'd relent, then the next morning was horrified when he saw I wasn't kidding. He wouldn't deign to take the bus, and he didn't have the guts to call the older girls down the street to hitch a ride. I told him he could walk (3 miles). He wound up asking a friend's father to drive him in - not my preference, but he only did it once and knew he couldn't impose again. He spent the day working on an apology and a better attitude.
It's okay to let them be "grown up and independent" and then give them the responsibilities that go with that! Usually they back down because they want rides, allowances, sleepovers, or homework help.
The thing for you to do is to realize that you still hold all the cards, you just have to change what you say and how you say it. And she is not smart enough to know who's the worst mother in the world! Don't believe her! Isn't there a book with a title something like "I hate you. Can you drive me to the mall"?? Get it from the library and read up! They ALL do what your daughter is doing. Just take a breath and a step back, try to be amused by her, imagine that she'd last about a day and a half in a home run by Mrs. XXX the perfect mom (who might have some different rules but probably is just as tough or worse in a number of areas). Take the drama out of YOUR thinking, and it will defuse the drama in your daughter's.