This is the delight of a journey to share knowlege with those who are thirsty of love ,fear and hope for Allah Azza wa Jal.May this be an humble quest for all of an quenching thirst for knowlege,and an endless request of help,understanding and Hidaya Taufiq from Allah.O treasurer of Knowlege may you all live the Blessing of this command 'Iqra !
iqra bismi rabbika ladhi khalaq...'
If the adoption of that girl was of the first type, whereby this girl was regarded as being the same as the couple’s other children in terms of lineage, inheritance and mahram relationship, then it is haraam and it is not permissible to continue it, whether the wife has died or not. If your wife did that during her lifetime, it is not permissible to continue that after she dies.
But if it was of the second type, there is nothing wrong with it; rather it comes under the heading of favours and kindness to other people. If the adopted girl was also an orphan, the Prophet (blessings and peace of Allah be upon him) said: “I and the one who sponsors an orphan will be like this in Paradise,” and he gestured with his index finger and middle finger. Narrated by al-Bukhaari (6005)
So carry on with what you are doing of taking care of this girl and sponsoring her, without attributing her to yourself or making her an heir like one of your own daughters, or regarding her as one of your mahrams who can uncover in front of you, or with whom you can be alone, or who can travel with you like one of your own daughters, once she reaches puberty or is close to it, and becomes a woman who is regarded as desirable. All of that is not permissible.
There is nothing wrong with you marrying her, if that is appropriate, or with one of your sons marrying her, if you have a son.
Please see the answers to questions no. 5201 and 185184