I've been teaching the piano to kids 4 and up for the past 15 years. This has happened to me before, and I tried hard to coax, reprimand and even bribe my timid pupils, only to realize that something was amiss in the first place- RAPPORT.
This is very important when working with kids, especially those who are forced to take piano lessons by their parents. Establishing rapport doesn't only mean that you befriend your kid. Be many things to him- a mother/father, a best friend, a playmate and a teacher.
An 11 year old girl is not that difficult to handle- just remember the things you were interested in when you were that young- boys, friends, and fun.
Make your lessons interesting, be lively, don't just sit while counting the beat-it's too boring for her. Mind you, they love to hear stories from your childhood piano days! They feel your empathy. Teaching the piano is a private affair- so you have to tailor your lessons according to your pupil's needs. Don't just conduct piano lessons- be a piano teacher!