Hello. I believe I can be of help to you with this issue.
As you know, anorexia nervosa
is a fairly chronic, relapsing disorder. I would hope that the treatment team ask to meet for a couple of family sessions to discuss and work out a plan for the daughter with anorexia to gradually eat comfortably without the sister present all the time. You mentioned that they wanted to do this step by step but it went nowhere. What was tried and what happened (if you know)?
There are no high schools I know of that treat eating disorders. However, there are some very good residential facilities where girls live in and continue their studies e.g., Avalon Hills ,Center for Change, both in Utah.
Relapses occur because the patient very gradually stops doing things psychologically that they learned to cope with stress
and the myriad of emotional issues. Their individual therapy sessions may plateau or stagnate and most importantly, they begin to regulate emotions and stress through the maladaptive thinking and behavior patterns they fell into when they became anorexic. The same general principles of relapse are present in eating disorders as they are in drug/alcohol relapse, weight loss relapse, etc. Most simply, the patients stop doing the things that helped them begin to recover and when faced with new stress in their life, they gradually begin to fall back into old patterns of thinking and coping. Dealing with relapse is actually not a complex,new treatment, but a very patient, deliberate re visitation of what helped the patient recover, examining how they abandoned that behavior, ways of coping with stress and negative emotion, etc. The fact that the girl with anorexia appeared to be on a solid recovery path is a very good sign. Usually, it doesn't take as long to get them back on track when they relapse, if they have at least one good experience with a "recovery trajectory".
Let me pause here because I could ramble on and on; I want your feedback to make sure I'm answering your question.