What is the ruling regarding an unmarried sister communicating via telephone with a married man who has children, when she claims it is a "pure relationship" like that of a daughter with her father, and what should I do about it, given that she promised me she would not resume this communication?

It is not permissible for your sister to address non-mahram men unless there is a strong, compelling reason, with the assurance that no fitnah (temptation/discord) will arise, and without speaking in a soft, alluring manner. Foreign (non-mahram) relationships often lead to undesirable consequences. You must prevent her from doing so and advise her with wisdom. Everyone should remember the Devil's insidious approaches and his temptation of the children of Adam.