A girl got her menses at the age of eleven, so is she obligated to fast despite her poor health, and what is incumbent upon her if she is unable to fast?

If the reality is as you described, then fasting becomes obligatory for her, because menstruation is one of the signs of puberty for women, provided it occurs when she is nine years old or more. If she is able to fast, she is obligated to perform it at its proper time. If she is unable, or experiences severe hardship from it, she may break her fast, and she is then obligated to make up the days she missed when she is able to do so.