There has been a lot of good advice given here.
I specifically loved Sailor's comments about how his own step-father handled his bad behavior. I do this with my own step-children AND children. I don't let them verbally abuse me. When they say or do something hurtful, I gently tell them that they've hurt my feelings. All of our children have good hearts and they're all defenseless in the face of learning that they've hurt someone.
I do not punish my step-children, nor do I insist that DH does. I just keep on doing my thing with them: I model the kind of behavior I would like to see from them and I'm never shy about letting them know how their words/actions effect me (positive AND negative).
That said, I do sometimes punish my own children when their behavior is particularly hurtful or inappropriate. For example, my DD (10 and learning the meaning of "hormone") recently called her sister an as*hole. I consider myself an AP parent, but in my world, calling your sister an as*hole warrants a consequence. DD knows better. She was just being mean for the sake of being mean. She's not 3. She's old enough and skilled enough to be able to be expected to handle a difficult situation in a more appropriate manner. And we have a zero tolerance policy for cruelty in our home.
I've digressed, but what I really wanted to do by posting to this thread was to validate YOUR feelings as the stepparent.
OP, it's okay and appropriate to be confused, jealous, angry, at your wits end, and downright pissed over being mistreated. You came here and you asked for support. And I think that's okay. I also think it's okay for you to think your step-daughter is acting like a spoiled brat. I don't think it's okay for you to act that way toward her, but if you feel that way, I think you're entitled to have a community in which you can safely talk about your experience as a step-parent.
This is me giving you support. It's really hard to be a step-parent. It can be lonely and frustrating. You can't be expected to muster the same feelings for your stepchild that you have for your biological children. If you ever achieve that, then I applaud you. But don't let anyone tell you that you're a bad step-parent if you don't love your step-children the same way you love your own children.
The transition to a blended family can be as stressful on the adults as it is on the children. And just as the children need an outlet for the feelings they encounter through that transition, so do the adults.
You're a good person. You care about your stepdaughter. I can see that. Don't take to heart the criticism of some of the poster's here regarding how you feel. You're entitled to your feelings too!
You can support your step-daughter by allowing her to be angry, by helping her to find more appropriate ways to express her anger, by letting her know when her words/behaviors hurt you, by modeling the kind of behavior you want to see in her, by reaching out to her in her calmer moments and trying to be her friend, by asking her from time to time how she's feeling and then just listening when she responds, and by continuing to love her mom.
This is getting long, but let me just say one more thing: you are not ruining her life, no matter what she says. You are not responsible for her pain. It can be very easy for step-parents to internalize their step-children's pain by convincing themselves that "maybe if I hadn't married her mom, she would be a happier person."
Be gentle with yourself and give yourself the permission to feel pain and anger, too.
It's really okay.