Call the mother....speak with the principal ans school psychologist/guidance counselor. Put it papers for a Pupil Support Team meeting...It sounds to me that this little girl has issues that need to be dealt with by a professional and possibly be placed in a special class for children with emotional issues/behavior problems...I know I am jumping the gun with saying this, but please start with referring this child to the Support team in your building for possible testing.
I think as she follows you around she has connected with you and finds you are sympathetic have you spoken to her Mom about her behavior as there could be a problem there, it is good she has communication with you .,even to talk about how ill she feels.,
Tess, have you gone to your Principal with your concerns? Someone from admin needs to sit in on your classroom, and evaluate her for further intervention. A child like this is disruptive to everyone in the classroom, and her behavior isn't a reflection on your ability to manage the classroom. Don't be embarrassed to ask for help, she really is exhibiting bizarre behavior.
Have you had several parent meetings about her disruptive behavior? This child's behavior is really beyond the pale.
Please don't continue to do this alone if you are right now. It's January, and her chances of adjusting normally to the first grade class are now gone. This isn't like a behavior problem that surfaces in early September.
Best wishes.
Could she have problems at home or misses her Mom to me it sounds like she is asking for more positive attention, do the children get plenty of outdoor activities or Fun and games when they are not learning? The fact she finds all kind of reasons like she is ill or hiccups, she is wanting attention and it is her way of getting it.