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.
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.