First, take a look at how you treat her. Are you always respectful with her? Do you talk to her in a similar manner that you would talk to an adult? I've watched many parents demanding respect from their child while being disrespectful themselves. Do you ask her to not interrupt when you're talking or do you order her to stop? for example. Your tone of voice is very important. Its so easy to slip into a disrespectful tone of voice when we're frustrated.
I'm sure your frustration level is high and it shows in your actions and tone of voice. Find a way to regain control of yourself so that you're not reacting to her in frustration. You must show her in your tone of voice and in your actions that you are in charge. It's a subtle but very important difference. When you react in frustration you're modeling for her that when one is frustrated it's acceptable to be disrespectful.
I suggest that you immediately intervene with an immediate consequence whenever she talks back or nudges you out of the way or hitting you, by immediately sending her to her room. Use a calm, unemotional tone of voice. Tell her she can come out when she's able to say she's sorry for being disrespectful and can act respectfully.
Talk to her ahead of time about this new way of discipline. Explain to her that this is not punishment. It is a way to allow her to regain control of herself and a way for you to maintain control of yourself. Perhaps even tell her you're in this together; that it's your responsibility to help her learn how to be respectful. Be sure to let her sisters know that this is the plan.
Just this week my 7 yo was telling me he was not going to bed. I left the room and when I came back, he was in bed and told me he was sorry but I started it. He has a speech disorder and I thought perhaps I'd misunderstood and so I just looked at him puzzled and asked what he'd said. He said, "we're in this together?" I chuckled, picturing how so very much we were in it together and said, "Yes." He then again said he was sorry and snuggled up next to me. The fight was over.
At first you probably will have to stop what you're doing and literally stand firm and repeat that she is to go to her room until she's able to tell you she's sorry and be respectful. BTW how long she stays in her room is up to her. Don't physically force her to her room. Just do the stare down.
With some kids you can take their hand and lead them to their room. If your daughter will follow and you can maintain your cool, try this.
The idea is to immediately send her to her room. She knows she's been disrespectful and you've reinforced it by telling her she's to go to her room until etc. Don't try to teach or explain with words. You've already tried that and it isn't working. Do not allow any space for anyone to continue a conversation or get involved.
If she doesn't say she's sorry or she says it in a sarcastic way, she immediately goes back to her room. No discussion. Remember you are showing her that you are in control not only of whether or not she goes to her room but also of your own emotions. Be matter of fact. It might help for you to think of pleasant things even while you're being firm with her. Definitely do not go over and over in your mind that you are going to make her behave.
Whether or not she behaves is her decision. She knows that when she chooses to not behave she will be in her room until she chooses to behave. You are in charge of what you will allow in your presence and she chooses how she will be in your presence. If she chooses an attitude or action that you won't allow in your presence then she has to out of your presence in a space so that she can think about her choice.
My daughter allows her children, 10 and 7, to play in their room if they want to do so on the theory that we often need a break in the way we're thinking in order to make a better decision. Even tho they're playing they are working thru the issue in the back of their mind.
Remember, the choice to be respectful or not is her choice. You're sad when she chooses disrespect. You're respecting yourself by requiring that she go to her room when she's behaving this way. You're pleased when she chooses respect. Give her praise. Along the way tell her you know that she will make a good decision.
Don't take her out in public until she can be respectful at home. Let her know this in a respectful way. Tell her you're hurt and embarrassed when she's disrespectful. If she responds with a so what, that is more disrespect. Stop talking with her and send her to her room. This reminds me: always make your statements based on describing her effect on you. Avoid blaming her. Just make a statement about your expectations that she be respectful. "I expect both of us to show respect to the other one." "When you talk back, I feel angry and want to yell at you. I'll tell you to go to your room so that both of us can rethink this issue."
You are in it together. Remember that lecture and explanations don't work. You're to the expect good behavior phase because she already knows the details of and the reasons why you expect what you expect.