Im sorry, but you dont sound like you care for this child very much
She just went through more than any kid should have to in a lifetime. She doesnt feel loved, she's angry (and rightfully so!) she probably has never been worked with or taught things (respect, appropriate behavior, morals, values, not to mention education!)
Is she seeing a counselor? If not, you need to get her one ASAP. Have her go alone and then you go with her.
It has been my experience that once a child feels loved, secure, builds trust, and is taught morals and respect, that they will show that in their behavior. If they were never taught not to lie, they simply dont know that its wrong. Children learn what they live.
Even if her parents (who obviously are not good parents) gave you permission, the LAST thing I would do is hit her, and if you think that spanking doesnt hurt, even by the smallest hand, I hate to break it to you, it does. All that is going to do is tear down that poor girls spirit even more at this point.
Talk to the school, find out if they can set her up with a tutor., someone who can find out what skills she has, and build upon that. Im thinking that she is acting out not only because she doesnt know any different, but because she is frustrated about not understanding things at school.
If all else fails... bring her here!