Is it permissible to pay off my debt to my grandmother (the lender) by undertaking her medical and personal expenses instead of returning the money in cash, and is it permissible to give my mother (who manages my grandmother's affairs) a sum from this debt in exchange for her management?

If you lend money to someone, it is not permissible to accept a gift or benefit from them before the debt is repaid, except in three cases: if it was customary between you before the loan, or if the lender intends to return a similar gift, or if they consider it part of the debt. Therefore, if you gave your grandmother something while she owes you money, and this was not your usual practice with her, she is not permitted to accept it unless she intends to return something similar or considers it part of her debt. If your grandmother is unaware of the ruling on accepting these gifts, you are not permitted to count this expenditure as part of her debt without her knowledge and consent, and you should inform her of it.

It should also be noted that the maintenance of a poor grandmother is obligatory upon her capable grandchildren if she does not have sufficient money for her needs, or if her closest relatives are poor.