Hi, E.! We have three kids, two of them girls, who we have gotten to the age of 16 (and past) with RELATIVE success ~ they are by no means perfect but they are good kids. We also have three little ones "coming up" behind them, and I have worked with youth for the last seven years. I know exactly what you're talking about, and of course it's completely normal.
Respect is key, in two ways:
FIRST, you need to model the respect you're looking for. Just because she's a kid doesn't mean she's not worthy of it, so do what you reasonably can to show her respect. Respect her boundaries, privacy, and space, again within reason. Listen to what she has to say as much as you can and try your best not to judge or attack, or she'll very likely shut down, and BELIEVE ME you want the lines of communication to stay open! Speak to her calmly. Validate her whenever you can, and whenever possible let her come to her own conclusions and work out her own solutions ~ then let her experience the consequences of her choices. Show interest in what's going on with her, and spend time with her. Teens know instinctively if an adult values and respects them, and it goes a LOOOONG way!
Second, and to balance out what I've just said, you need to insist that she return that respect. That's where the above becomes invaluable, because you can point out when and how you've tried to respect HER. There are certain things that you should probably ignore, like if she turns up her nose at the food you've cooked or pouts about having to go to Grandma's instead of Susie's house (i.e. choose your battles very carefully)! But when she does not speak to you respectfully, call her on it and walk away from the conversation until she is ready to do so. Insist that she respect the family rules (make those rules few and reasonable) and that she pitch in to do her share. Be clear and consistent with discipline and FOLLOW THROUGH ~ she NEEDS you to be predictable! If she is hurtful or destructive to ANYONE, you should demand that she make appropriate amends. Finally, watch carefully for signs of disrespecting herself (alcohol, drugs, sex, etc) and be ready to intervene.
I've had many, many "lows" as a parent, but one of the highs (this WILL make sense when you get to the end!) was when I found out my 15-year-old daughter had been going out to the high school parking lot to try smoking pot with a "friend." I confronted her very calmly, simply looking her in the eye and saying, "Is this really who you are?" She broke down and spilled a lot of stuff that had been going on with her, and it was one of the best conversations we ever had. She hasn't done it again since (that I'm aware of) and pretty much from that point on her grades and attitude gradually improved, and she is now a successful college student.
Okay, I'll shut up now! Best of luck!