What you said
You've exhausted your strategies without achieving the desired result. This one girl will not comply with you until the negative consequences of her poor behaviour outweigh any benefit to her. Look again at the strategies that you find distasteful - maybe she needs to be humiliated in front of her friends, maybe a raised voice will make her think about challenging your status. Remember, "If you always do what you've always done, you'll always get what you've always got". Henry Ford said this and it's true.
The expert view
Stop trying to rationalise behaviour that is instinctively emotional and selfish. She doesn't need understanding. She takes your sincerity and attempts to reconcile as a sign of weakness. You don't convince someone to be kind - you give them reasons not to be unkind. Simply apply the school sanction system every time she mugs you off and accept that not every kid will bend to your will. You're new to them, so as far as they're concerned you're a supply teacher. Your relationship with the class will take a while to build up but for now you're sport. You'll get them in time but you need to go structural with the strategies and use interpersonal tactics later on when they come round to you a bit more.
Tom Bennett's latest book, Teacher, is out now. Post your questions at www.tes.co.ukbehaviour.