Bird I'll begin by saying I do not envy you the task you've undertaken. From what I've read of your writings I presume you grew up in a more or less standard family, and that does NOT equip you to deal with a System Kid. Make no mistake T is a System Kid, she has learned to work the system for everything, and at age 11 she's probably both smart and very close to a degree in manipulation.
I say this for a few reasons, one being I grew up in a kid prison called an orphan home, and I have tried to help some system kids get their heads screwed on right. I've also spent enough time in Schools to know America's Public Schools have become near useless.
T has learned by now to set her goals for the day, scam somebody into providing what she wants, and if the first target doesn't deliver to move on. T is also quite probably at the point of having detached from any familial relationship. To System Kids everything every day is another deal. What have I got that you want, and what have you got to trade is a far more normal thought than let me share the cookies with you.
Your description also indicates T has developed some skill at pushing NO. Kids need somebody who says NO, it keeps a lot of kids from getting killed or jailed, and in T's head no is a starting point. She probably accepts NO from nobody, and isn't likely to from you until she values associating with you sufficiently to not risk being sent off.
Sometimes kids like T can grasp the concept of relating to examples. One I've used a few times with success is the movie/concert example. You need a ticket to get into either, and without a ticket you stand on the sidewalk watching other people go in. Think of school as the ticket to life.
Given T's age the primary thing you need to do with her is be honest. T has one hell of an onboard lie detector, and kids who know the system operate from the point of everything being a lie until proven. You must deliver on every promise or you're just another liar. Be careful what you say. Promise nothing instead use phrases like I'll try.
Most important always listen and do your best to hear without reading into what you heard. on't assign tasks, but ask for T's help. If she values associating with you she'll help. If not, leave her to solve her own next problem by herself. Remember everything is a swap with her.
For your own protection maintain a journal of all interactions with T. If you're lucky it won't ever be needed, but if the sh!t hits the fan a journal can save you from a Social Worker, and you have no idea of the stories a System Kid can whip up. I well recall a 12 year old darling who went to school MONDAY with bruises from combat badminton on Saturday with her foster siblings. She decided it would suit her purpose to tell the teacher her parents beat her. On the way to the office to assist the Principal write the official report she decided it was a good idea to claim the parents were starving her too. When the Principal called mom and looked in the lunchbox the little darling was carrying to confirm mom's inventory Miss 12 year old genius hit the skids. From then on Miss Genius had to prove every allegation she made.