There are numerous reasons why this child may be acting out. I'll list out some of them with links you can click to discover a little bit about what may be going on.
To provide perspective, I have experience with some of these issues, but only a licensed medical professional can make the call to determine if your child has any of these. To be clear, there is something wrong, and although it is always in part the parents' fault1, there may be more at play that requires a closer look.
I'll go over some of the ways I found effective for managing both ADHD and ODD, and what my parents did to facilitate my growth.
Just so we're clear, I am not a licensed medical professional, nor do I play one on TV. Please take all advice on a Q&A site with a grain of salt, and seek a trained medical professional for diagnosis and advice.
I don't know exactly why your daughter does what she does, and she may not even know. What I will do is my best to provide some perspective, and to help you deal with some of the issues she may be facing. It is up to act on them.
So to start, my first thought immediately went to an Adjustment Disorder. With these, it usually happens when a person can't cope with the stresses that come with a major life change. You mentioned that neither parent is in the picture, she's been shipped off to multiple places, and my guess is that she doesn't feel like she really has a home. Think of it like nationalism. If you don't think you have a home, why do anything for someone else's? I'm not condoning the behaviour, but it's important for you to try to understand things from her position. She's 11, and has been for all intents and purposes left behind by the people who are supposed to love her the most.
It seems to me like you try to force her to do things, but try and be patient with her. I don't recommend this usually, but if she wants to run out into the street and cry wolf, tell her what will happen if she does. The important thing is to let her know that she's right and you don't make her decisions for her, and that it is up to her how she wants her life to go. From my perspective, you came to the table expecting a battle, and my biggest question is "Why?". This is an 11 year old you're fighting, and she's old enough to understand words. Talk with her. Have her understand you're not going to be another change in her life, and explain to her that you're here to stay. Be an anchor she can trust not to give her up again, and keep in mind that trust is earned.
I personally have struggled with this, and some of what she may be struggling through is frustration based. ADHD is actually a developmental disorder with stunted brain growth. It doesn't mean that she's dumb, mentally challenged, or (to use a personal favorite with my school bullies) a retard. It means that her brain maturity will develop slower than her peers, and that may cause some serious frustration. Do you ever see her focusing intently on a single task, or constantly fidgeting? These are classic signs of ADHD, and the frustration when the focus (that we have so little of) is broken often makes people who have it lash out.
The best way to counteract this is to provide constant encouragement, and above all consistency. If you're not consistent in your punishments when she lashes out, whether due to frustration or a break in focus, it can cause confusion and further frustration. It is up to you to break the cycle and provide her with the support system she needs to succeed. To be honest and provide a critique, you used the word battle when describing your situation, and I personally don't think that's the right way to go about it. You're her guardian, so it's your responsibility to work WITH her to get through these problems. Remind her that you're on her side, and change things together.
As a note, if ADHD is indeed the culprit, please watch for signs of substance abuse and addiction. Especially at that tender age, drugs like speed and cocaine are not only hugely addicting, but provide the exact release for people with ADHD. Having suffered from these myself, they are incredibly harmful, and were a large part of the reason for my constant negative attitude towards my parents. It is imperative you make sure that help is obtained before substance abuse becomes an issue. Drugs are a seriously slippery slope and I cannot stress enough the important of avoiding them especially in belligerent children who have ADHD.
Oppositional Defiant Disorder is particularly nasty, and when paired with ADHD can be a recipe for absolute disaster. Normally, parents with children who have ODD often believe their child is just an intentionally bad child, or that their parenting caused such issues with them. It is often misdiagnosed or downplayed, and can be dangerous not only to those who interact with the child, but also to the child themselves. To clarify, CD is ODD but with intent to harm, and usually attempts to violate the basic human rights of others. Things like hurting animals, intentionally injuring people, or not seeing why it's an issue are some signs that it may be CD rather than ODD. Both of these are exacerbated by children with ADHD, as ADHD causes anger with ODD, which causes frustration triggered by ADHD, etc, etc. It is a constant mental struggle, and requires a medical diagnosis by a licensed professional.
Take heart! I never knew why I acted out and it wasn't until I was 14 that I was formally diagnosed with ODD. The best way I've found to handle it is patience from my parents. My best advice for this is to let your child know that you are there for them when they're ready to discuss things peacefully. An accurate analogy that I've found is the modern day second amendment. No this is not meant to poke any political bears and yes I heard this from a comedian, but it is as accurate as I can find for the feeling. The second amendment doesn't make as much sense as it used to because it used to take a while to load a musket. You couldn't just shoot someone on a whim when they angered you, and it gave you time to reflect on why you were actually angry. By the time you had it loaded, you realized that maybe not all was as it seemed upon first glance, and it gave you a chance to stop before you did something you may have regretted. The same principle applies with ODD/CD. As with all of my advice, talk with her calmly. Ask her to next time she's angry at you to take 60 seconds, and if she's still angry after 60 seconds you can work on it together. The key to this is working with her rather than against her. The more you work against her, the more her ODD/CD will take over and she will be for lack of a better term, oppositional.
I will admit, I don't have a lot of anecdotal experience with this (the closest I have is my cat when my girlfriend leaves the house but it's not quite the same). As I will constantly say, consult a medical professional to determine the best direction to head with these disorders. I can however help with solving some of the issues that are associated with separation anxiety. Similar to Adjustment Disorder [see above], separation anxiety occurs when a major change has been seen in a child's life, specifically with this disorder involving separation from someone they are close to. To use a well worded quote from the APA:
"...Separation anxiety disorder is an excessive display of fear and distress when faced with situations of separation from the home or from a specific attachment figure."
Sound familiar? From your scenario, it seems to me that she may have some severe fear regarding being separated from attention, as if she will be forgotten or once again left behind. Again, similar to AD, I suggest providing a safe place for her, and constant reassurance that you're not going anywhere. She's a child, and fear brings out some of humanity's worst traits and sometimes can trigger a fight or flight response. It's possible she may be in a continuous "fight" state, and everyone is the enemy. Unfortunately, that's the best advice I can provide, but again, speak to a medical professional for diagnosis and further ways to help deal with it.
The jack of all solutions
Love and patience. It sounds simple, but in practice it's one of the absolute hardest things you can offer a child struggling with these or really anything. Right now, you have a chance to become the most important person in this child's life. Love and patience are the literal most valuable commodity you have to offer, especially for a child who has from what I can see been forgotten. Do not make the mistake of thinking it will be easy. Regardless of if she has any sort of disorder or not, she will fight you every step of the way and in many cases will not reciprocate the feeling.
What I can say though is you have accepted the responsibility to take her in despite her issues and problems. You have already started your journey to healing and it is not a bottomless pit you're dumping your love and affection into. There is an end to the hard road you face, and from personal experience and anecdotal experience from MY parents, it's supposedly the most rewarding feeling you will have ever felt.
But don't just take my word for it, give it a try. Use all of your patience, understanding, and love. I promise you will not be disappointed. I can't provide you with a medical guarantee though, because well....I'm not a doctor.
1Whether it's through lack of understanding, perspective, ignorance, confusion, intention or anything else.
2Please be wary that if CD is diagnosed to be the issue, it can rapidly devolve into ASPD, or Antisocial Personality Disorder. Without treatment, it can involve physical abuse to any number of things. It is not diagnosed until the age of 18, but please be aware that it is a very real and incredibly troubling issue.