Dear A.,
As a parent of an eleven year old, nonverbal autistic boy, I decided to put in my two cents. Not to give you false hope, but my son hardly said a word until he was nine years old. While his vocabulary has vastly improved with the PECS system, sign language usage, and therapy both in school and private sessions, he is still considered to be nonverbal. If you or the girl's parents have not already implemented a Picture Exchange System (PECS), then do so. It has helped my son communicate what he wants.
Now the hard part, autistic or not, she should be disciplined. In fact, the stricter you are, the better for her in the long run. Remember when it comes to punishment, five minutes is like an hour. What this means is that when you ask her to pick up a toy, you give her a warning to pick up her toy or no computer. You don't have to pick the computer, but think of something that she enjoys and take it away from her. If she picks the toy up, great. However, if she does not pick it, then punish her by taking away (fill in the blank). Here is the tricky part, she will get mad. She will scream. She will cry. You will feel bad. You will want to give her back her (fill in the blank). You may even feel guilty enough to pick up the toy yourself. Don't. What you are doing is a battle of the wills. No matter how long it takes, you must win that first battle. It will take a few times, but once she realizes that if she does not listen, she will lose her privileges, she will actually listen. If she is like my son, she will learn who will cave in to her and who will not.
Another thing I have used with my son is the first/then technique. One scenario is when he wants ice cream, but he hasn't finished his dinner. There are lots of times when that happens. What I tell him is either first dinner, then ice cream. Or I will tell him, If you want some ice cream, eat your dinner first. Again with first dinner, then ice cream. He doesn't have to eat all of his dinner, especially if he is given a giant plate. He does have to eat at least half of it. He eats his dinner pretty quick once he realizes that he gets his ice cream afterwards. Now, like most boys his age, his mouth is a black hole that leads to the bottomless pit known as his stomach.
Like all kids, you can't expect them to be perfect. She will slip up now and then. However, stay persistent. Keep at it. I don't expect her to be like my son. When another classmate makes a mess, he cleans it up. One day you might be surprised. Remember this, she knows exactly what is going on around her. Just because she is not talking, it does not mean that she is not listening to what is being said.
As for the brushing of the hair, have you tried a wide tooth comb? Different bristles. Maybe even take her to the store and let her pick her one brush or comb. One that only she can use, that is all hers. Enough with this novel of a post. I hope I could help out in some way. Good luck!