Is it permissible to send Zakat money to a Christian orphaned niece living abroad, whose mother abandoned her, with the intention of helping her and inclining her heart towards Islam?

It is not permissible to give zakat to your disbelieving niece merely because she wishes to come to you, because inclining her heart towards Islam is not a justification for giving her zakat money. This is due to the absence of the condition stipulated by scholars for those disbelievers whose hearts are to be inclined (towards Islam) from zakat money, which is the Muslims' need for that. The Shafi'is do not permit giving zakat to a disbeliever at all. As for the Hanbalis and Malikis, they permit giving zakat to a disbeliever if they are among those whose hearts are to be inclined, meaning they are leaders of their people whose conversion to Islam is hoped for, or whose evil is to be averted.

You must seek assistance for her from another source other than zakat, and strive to bring her over and invite her to return to Islam, because she was a Muslim by virtue of her Muslim father. So, if a person's father is Muslim and their mother is Christian, then they are considered Muslim before reaching puberty, following their father. If they reach puberty and choose a religion other than Islam, that is an apostasy from their religion.