I went through a very similar situation where my step daughter's mom abandoned her. She is still trying to recover and she's 27. She wants her mother's love, appreciation and approval and doesn't get it. She didn't get it growing up and she's still trying to please her mom and getting the same treatment. I've tried many times to explain to her(since she's been an adult) that I don't think her mother's capable of the relationship she's desiring. In her mind, I know she knows that. Her heart is a completely different story. It is so sad to watch. She knows she is loved by her Dad and I, but she wants it from her mom.
Having made lots of mistakes I have a few ideas. A GREAT book is Love and Logic with Teens. It is an easy read and even if you decide not to implement all of their ideas, it will give you ways to deal with issues without becoming emotionally drawn in. It also helps you see how to have the child "own" their mistakes.
If you can afford or have access to counseling, I would do it. We had counseling for our daughter, but she wound up going back to live with her mother at 14. Her mother abandoned her from 9 until then and when her mother told her she "wanted her back" she was thrilled. Again, because she had abandoned her in the first place (and her mother also told her she wouldn't have any rules and what kid of 14 doesn't think that's a grand idea). The things your step daughter deals with are hard. Being her age is hard too. It's a difficult time under normal circumstances. She doesn't know why her Mom doesn't want anything to do with her and she hurts because I'm sure most of her friends have their mother actively in their lives.
Having time out with just she and you would be good too. Maybe grab a coffee/hot chocolate together. Talk when things aren't heated. Ask her what her dreams are. Maybe talk about volunteering activities she can do.
Another thing I tried to do because my step daughter was so damaged in her self esteem was before she went to bed every night I tried to compliment her about some part of her character. Example: You were so patient with your sister when you helped her find her shoes. I really appreciate you helping her out like that. Another idea my pastor talks about is he started a notebook for each one of his kids. Through the week he tries to write them a love letter once a week. Sometimes he shares wisdom, sometimes it was more like you really tickled me when you...etc. When he wrote something, he would place the notebook on their bed while they were at school. After they read it, they gave the notebooks back to him. He saved all the notebooks and when they headed off to college, he gave them to his girls. I really think that's an awesome idea. I haven't started it, but I plan to.
I don't have any great answers. I've been where you are and it's hard. Parenting is such a hard job and being a step parent is hard as well.
Blessings!
L.