From what I am learning with my DS6, each child is different, and what might be right for one child might not fit another. Does she like school? Is she happy there? Is her school willing to work with you?
There are a number of private gifted schools in the Chicago area. However, just because they are a gifted school doesn't mean they would be a fit for your daughter.
I wouldn't immediately assume that she can't get an appropriate education at your local public school. No matter where she attends school you will need to advocate for her. The first step is in talking to her current school and seeing what they can do for her.
