Is it permissible to give my Zakat al-Mal to my two orphaned nieces, whom I sometimes support financially?

If the maintenance of your brother's two daughters is obligatory upon you, then it is not permissible for you to pay zakat (definition) to them. However, if their maintenance is not obligatory upon you, then it is permissible to pay zakat to them if they are poor. The maintenance of a poor orphan falls upon their wealthy heirs, based on the Almighty's saying: "And upon the heir is the like of that." So, if you are their heir and capable of providing for their maintenance, it becomes obligatory upon you, and it is not permissible to pay zakat to them. But if you are not an heir, or are incapable of providing maintenance, then it is permissible to pay zakat to them.