Wow. Just calm down.
She's eleven. She keeps herself clean. Who cares if her hair sticks up like a cartoon character, or if it's purple or whatever.
Your characterization of her looking like a "bum" is harsh and cruel. A homeless person is often trying to survive, to find food, to find shelter, and I would imagine that hairstyling is the least of his or her concerns.
I would find another way to think about this. Your daughter's hair is uncombed, but it's clean. WHO CARES? As she gets a little older, she may develop a desire to style her hair, or she might buzz it off to keep things easier, and if she's asking about an easy-to-keep style, you might treat her to a salon appointment, as long as you stay out of it and let her get it super short if she wants.
You've obviously taught her to keep her hygiene in check. Good job. Now let her have some independence and wear hair that would make a porcupine proud. She sounds like she's carefree and not worried about what people think (to a degree - she makes sure she smells good and is clean, but if she's got crazy hair, that's fine with her and it's their problem if they don't like it).
Apologize to her for the boot-camp march to the mirror, and tell her that you're pleased that she understands the difference between good hygiene and styling. Lovingly ruffle her fun hair in the morning as she confidently goes out the door to school, give her a hug, and count your blessings.
I can't imagine telling a therapist that your daughter showers, brushes her teeth, uses deodorant, but has unruly hair, so should she be in therapy? There are no in-patient psych hospitals for unkempt hair styles.