I think talking with a pediatric psychiatrist would help a lot. They could diagnose her and give you tips on when to give in, and when to make her do something. Many psychiatric hospital here (in Canada) have youth wards that offer some outpatient services. Because she is probably pretty freaked out about things and confused herself, having someone to confide in may be helpful. And she may be able to connect with other kids having some of the same problems she is. That is powerful. And you can connect with parent's struggling too. Nothing like learning from others who have been there.
It is tough. I had a nephew with mental health issues and he blamed many bad behaviors on that and used the 'I can't help it' line when he could help it. When he did have control and was pitching a fit to get what he wanted. I always knew the difference because once he got what he wanted he would smile and settle down. A person with bipolar disorder doesn't just all of a sudden become calm because they got what they've wanted. His parent's didn't seem to be able to tell he was manipulating them. But when he was in an episode you could tell, his eyes would get a different look and nothing would calm him down until he wore himself out. The depressions were harder in many ways, so tough to see someone so young so despondent.
So please get professional help. I wasn't 11, I was about 14 when I started having problems with depression, and I didn't tell anyone. She is very lucky you notice and will get her some help.
You are a good Dad for wanting to get her help. Be it bipolar or something else, you know something isn't right and that deserves to be evaluated by a pediatric psychiatrist.
And if she is bipolar there is so much reason for hope. They are coming out with better treatments, and learning more and more about behaviours to help manage bipolar like keeping a predictable schedule and exercise. If she is bipolar - chances are she'll be okay. The road may be steep, not saying being bipolar is easy but you can have a really good life. Kay Redfield Jamison is one story of success, so is Patty Duke, and so many other people who, with treatment, are living normal lives.
I think it sounds like it could be bipolar. I think the best thing to do would be to go to a doctor with these issues. If you aready are talk to them about your concerns with your daughter. I might also suggest you go to therapy to help you learn how best to deal with your daughters behavior, or maybe family therapy. I hope you get it all figured out. Best of luck to you.