Is a divorced sister, who is supported by her father of limited income, and who has no source of livelihood other than what remains of her ex-husband's money for her children, eligible to receive Zakat?

It is permissible to give Zakat (definition) al-Mal to your sister if she is in need, and her ex-husband does not provide for her, and your father's provision is insufficient for her and her children. She has a greater right to it than someone who is not a relative. Giving charity to her is better because it combines charity and maintaining family ties, as the Prophet (peace be upon him) said: "Charity given to the poor is a single charity, but charity given to a relative is two: charity and maintaining family ties."