Your post leaps all over the place, to be frank, and it's hard to tell where to start.
The psychiatrist recommended writing sentences as punishment? Seriously? That's what I assume from the last part of your post where you say "the psychiatrist said to make her finish the assignment." I find it very odd that the doctor would say that writing lines is an appropriate discipline for this particular child because it is NOT working. The psychiatrist should be working with the parents to find this girl's "currency" -- the thing she values so much that losing it will hurt her. That is the thing to take away from her when she behaves in negative ways. Making her write sentences is not working; why are they repeating something that is not affecting her enough to make her act differently? The parents need to meet with the psychiatrist without the girl there and talk seriously about different discipline.
Things like setting a fire and leaving the house with her brother as you describe are very serious safety issues. What happened after those incidents? How was she disciplined after the fire? How did she manage to start it and why was she able to get hold of matches or a lighter--?
Why, if she is saying "I lost the paper and pencils" is she then handed new paper and pencils? That teaches her nothing at all except that if she loses something (which possibly was a lie anyway) that she gets the same thing given to her.
Why is she eating in her room? Is the problem that she is expected to eat at the table but SHE grabs her food and runs to her room? Why then is she allowed to eat at all? No table, no food.
It sounds as if the parents are overwhelmed and overrun by this girl. I would bet that they "have tried everything" -- which means they have not stuck with any ONE discipline long enough for it to sink in with their daughter. They must, must get their own counseling so they can come up with a firm plan for how they will handle her--before she burns the house down.