What is the proper way to handle a sum of money left by a deceased sister with her [other] sister for the purpose of 'Umrah, knowing that she had permitted her to use it as a loan, and no one else knows about it?

The amount of money is a right of the heirs, to be divided among them according to Islamic law. It is not permissible to spend it on performing 'Umrah on her behalf merely because of her intention, as upon her death, the money became their property.

If the amount is still with you and you did not borrow it during her lifetime, then hand it over to them. It is not permissible for you to borrow it after her death.

However, if you borrowed it during her lifetime, then it is a debt owed by you to the heirs. If it is due immediately, then it must be handed over to them. If it is deferred, then it must be repaid when the term expires.