If I were the ex-employer - I won't let 阿婆入老人院 if I have a helper at home. I believe a good helper can take much much much much better care of the elderly than putting her in the elderly home - at least the helper can bring her out for a walk or fresh air, cook the food according to her personal appetite and need, help her in day to day care on 1:1 basis. Besides, the expenses for a nice elderly home would not be lower than that of the cost of a helper.
It does not make logical sense. Unless, 阿婆 needs some medical treatment that requires special equipment that would not be available at home.
濕疹 is something related to the immune system of your girl, it should not be the responsibility of the helper. It is not fair to her, though she is not a good performer.