I empathize with you and your bullied teen. No child should have to endure such treatment, especially in school, where the authorities are supposed to keep her safe from bullies.
Imagine what it must be like for your daughter to have to go to school and be harassed and tormented every day. No wonder she has headaches and stomach aches.
Do everything in your power to stand up for your daughter and fight the good fight. Document what happened at the school, get names of witnesses and call the principal and dean of students daily to find out when the teen bully who choked your daughter will be expelled.
If the local police refused to press charges, take this to the county sheriff's office. If you have witnesses (the teens who pulled the bully off your daughter), then they should be willing to press charges. If the county sheriff turns you away, then go to the FBI. In the very least, you should be able to file charges with the school police, which means that this bully will face charges in the juvenile court system. Call the DA's office and ask for their advice.
Continue to take this seriously. Your daughter's life and mental health could be in danger. Recently, a 15 year old teen killed herself after being bullied and harassed on a daily basis. Do not underestimate the effect this has on your child. If you have to get a variance and send her to another school, do whatever it takes.