I understand that you don't want your child to be a brat- but I don't think you want her to FEAR authority either. The whole rolling the eyes, having trouble with homework thing is pretty normal at that age.
It sounds to me like you've punished and taken things away, but you don't really have a system. It's just sort of random and one time you let her get away with it and another you don't... it's like having a toddler again or a dog- consistency is the key thing!
Also, of course you want her to be polite and well-behaved- but respect is a two way street and must be EARNED, by both kids and adults. You don't want her to not think and act for herself either.
What kind of behavior is she seeing at home? Remember, kids imitate what their family does. If she sees you and your husband yell or talk back to each other or act a certain way, she will imitate you. Or of course she might be seeing that behavior on TV and 'trying it out' in real life. Or picking it up from another kid. You need to figure out where this is coming from- all kids try acting out once in a while, but if she is doing this all the time, its coming from somewhere.
I think first you need her to be CRYSTAL clear what your expectations are and what will happen if she does not meet them. Make sure your husband is totally on the same page with you, so she can't play you off of each other.
1) Write down as many SPECIFICS as you can think of that are NOT acceptable ( eye rolling, sighing, foot stomping, 'snotty voice', etc) We talk to my son a lot about his ATTITUDE being just as important as whether or not he does something. For example, he gets an allowance for doing his chores- but he only gets the money if he also 'shows willing' and does them without being nagged about it and without sighing or grumping around or complaining!
2)Talk about tone of voice. If my son says something back to me under his breath or in a 'snotty' voice, I call him on it RIGHT AWAY. I repeat what he said to me in the exact same voice. Then I ask him " Do you think that was a nice way to talk to me? What were you trying to accomplish with that? If you don't like what I was saying, you can tell me so, but not in a rude voice. It is NOT acceptable for you to talk to me or anyone else in that tone of voice." ( BTW, I know this one sinks in because he HATES it when I imitate him like that, lol)
3) All kids can be bratty once in a while or say something they immediately wish had not come out of their mouths- just like the rest of us. Teach your daughter to THINK before she speaks! I tell my son this over and over and tell him to think about his words coming out of his mouth and hanging in the air after he says them- they don't just disappear and what you say has consequences!
4)Have a Family Meeting. Go over with your daughter what you except of her each day in terms of chores, homework, etc. Make it totally clear what things are absolute deal-breakers if she doesn't do them, or mouths back or whatever. Remember, as a parent, put out the big fires and step over the small ones! Again, all kids are going to drop the ball once in a while- no one, child or adult is perfect- but she is totally capable of knowing that some things are just not going to be tolerated.
Also at this meeting, ask HER what she thinks she ought to have to do, what rules are reasonable, etc. I am not saying you have to agree to everything, but compromise is a good skill and if she is willing to clean up the kitchen for 20 minutes of computer time or her favorite TV show, why not?
5) Now- consequences and rewards. I honestly believe that it is SUPER important to have both. We all have to do things we don't enjoy - but if you don't want to do your work, would you go if you didn't get paid? Of course not. Your daughter has no choice about having to DO her homework, but rewards and consequences can make it easier for her to make a GOOD choice about that.
I think with homework, you the parent really have to be on top of it. Email her teacher once a week to check in on her progress- and let her know you're doing so. Check to be sure she's done everything she was supposed to each evening. If she hasn't- then the first consequence is no dessert, no TV after dinner. If my son messes around too long not getting his work done, he has to set his alarm a half hour earlier for the next morning and get up and finish it before school.
But- if he does what he's supposed to, homework and chores, no complaining, no nagging from me, then he can have some ice cream and watch some TV before bedtime. Afterschool hang-outs (I am no longer allowed to call them playdates, lol) are also contingent on how much homework he has and if he gets it done in a timely fashion with NO complaining.
We will take away TV show and video game privileges for talking back, not doing homework or complaining about it too much. He KNOWS that is a possible consequence of his behavior and if he complains after it happens- then TWO nights with no privileges.
We do reward him with $10 per A on his grade card- which can be quite a haul if he does well. I don't see anything wrong with that kind of encouragement. His allowance is $10 every 2 weeks- if he has done his chores, he gets paid when we do. If not... then not.
The key thing is to decide what you can accept and what you can't. Then make sure your daughter knows exactly where you stand and what SPECIFIC punishment or rewards she will get based on her behavior.
but this is all pretty normal for my 10 year old and other friends of ours too. Just think, train her now and it will all be so much simpler when she is a teenager! Good luck!