Making up for missed days of fasting and fasting the vowed days

Q: I suffer from a cardiac disease and it is difficult for me to walk. A few years ago, I traveled to the USA for medical care. I began to recuperate after an operation. My sister was in good health. Having deep feelings for me, my sister said: "If my brother keeps getting well throughout the coming year, I vow to fast twenty-eight days." Prior to the new year, my sister was infected with a disease rendering her bedridden, thus she could not observe Sawm (Fast) for four or five years. Although the family has tried to treat her illness in Al-Madinah, Riyadh, Jeddah, and USA, she still suffers from its impact. (Part No. 23; Page No. 329) That is why the days of Sawm she missed have accumulated over the years: Firstly, the twenty-eight days she vowed to fast. Secondly, sixteen days she did not fast during Ramadan because of menstruation. Thirdly, three months of Ramadan, as her health condition did not allow her to fast for three years. Moreover, the family, as a temporary atonement, fed a needy person for each day of Ramadan on which my sister could not fast, hoping that she would regain her health soon and perform Sawm on her own. But she is still ill. In sha’a-Allah (if Allah wills) Ramadan is soon to come,so we want to know what the ruling is if my sister hires a woman whose faith she trusts, as a proxy, to fast on her behalf on the following days:1- The twenty-eight days my sister vowed to fast, but could not.2- The sixteen days she needs to make up for missed Sawm because of menstruation.3- The four months she needs to make up for the three months of Ramadan over the last three years, as well as the coming one.Please answer our question in writing, may Allah reward you with the best!