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 is His slave and Messenger.
First of all, it should be noted that adoption is forbidden and we have already clarified this in Fatwa 82371. It is not permissible for this girl to be traced back to other than her real father. Allaah Says (what means): {Call them (adopted sons) by (the names of) their fathers, that is more just with Allaah.}[Quran 33:5] It is her real father who should be her guardian in marrying her off and she should be introduced to her real parents so that she could fulfill her duty of being good and kind to them.
If a person proposes to a woman and each one of them likes the other, then it becomes forbidden for another man to propose to her. However, if the first man who proposed to her is rejected or that he gives permission (for the second man to propose to her) or that the matter is not known, then, in this case, it becomes permissible for the second man to propose to her as stated by the jurists .
However, it is better to make sure whether or not the engagement was accepted, if this is possible.
As regards sending messages to this girl or talking to her through the internet or any other means, then this is not permissible even if this is with the intention of making Da’wah to her (calling her to Allaah) and showing her the right path. This is because such contacts are a means of temptation and it is an obligation upon the Muslim to avoid all means that may lead to temptation.
Your intention of willing to rectify her could be done by seeking the help of some pious women to do this, and this is better in order to be safe regarding your religion. For more benefit on the dangers of chatting through the Internet, please refer to Fataawa 84874 and 132000.
It should be noted that if a Muslim man wants to marry, he should look for a righteous woman, but marrying a woman who is not righteous on the pretext that he will rectify her afterwards, then there is a risk in this as she might influence him and spoil him instead of him influencing her and rectifying her.
Allaah Knows best.