Is it permissible for him to adopt a girl whom his wife had adopted when she was alive?
Question

does an adopted father has the right to keep an adopted daughter after the foster mother has died ?
Praise be to Allah.We have previously explained that the word adoption, as customarily used by people, may mean two things:
1. Undertaking to raise a child and care for him, whilst attributing that adopted child to the adoptive family and regarding him as one of them.
2. Undertaking to raise a child and care for him, without changing his name (i.e., still attributing him to his original lineage).
The former was permissible at the beginning of Islam, then it was declared invalid by the Qur’an. The latter is permissible and it is prescribed in Islam.
See the answer to question no. 126003
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 
And Allah knows best.