Is it permissible for the Zakat payer to help his employed sister from his Zakat money to perform the Hajj, knowing that the cost of Hajj is greater than her financial capacity?

Hajj is not a valid expenditure for Zakat according to the soundest opinions of scholars, because "Fi Sabilillah" (in the way of Allah), when used unqualified, refers to Jihad. Therefore, do not give your sister Zakat money for Hajj. Rather, spend it on those in need from the other categories mentioned in His (the Most High) saying: (Indeed, [prescribed] charities are only for the poor and the needy...) If your sister is able, she performs Hajj; otherwise, it is waived for her until she is able.