All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His Slave and Messenger.
You have done well by admitting your mistake in talking to this girl on the phone and having an affectionate relationship with her. So, you have to repent to Allaah The Almighty and do not repeat that mistake. You have also done well to instruct her to go to an Islamic center to learn about Islam. Please see Fatwa 81356.
Undoubtedly, it is not permissible for a Muslim to marry that girl if she persisted in disbelief because Allaah The Almighty Forbade the Muslim from marrying the unbelieving woman, except from the People of the Book. Allaah The Almighty Says (what means): {And do not marry polytheistic women until they believe.}[Quran 2:223] If she embraced Islam, a Muslim man (you or anybody else) would be allowed to marry her. In that case, it would not be permissible that her father concludes her marriage contract, because a disbeliever has no authority over a Muslim woman. Allaah The Almighty Says (what means): {And never Will Allaah Give the disbelievers over the believers a way [to overcome them].}[Quran 4:141]
It seems that your marriage to this girl will entail many dangers as it seems that your family and hers live near one another. Thus, you have to think about the matter logically and away from emotions.
However, as long as her family members are unbelievers, if she embraces Islam she may be instructed to leave her family and marry a Muslim man away from them. If she does not have a Muslim matrimonial guardian, the Islamic judge or whoever acts on his behalf may conclude her marriage contract. If there is no Islamic judge nor anybody who acts on his behalf, she may authorize a Muslim man to conclude her marriage. For further information, see Fataawa 88488 and 105117.
Allaah Knows best.