What is the Islamic ruling regarding the presence of an orphaned girl who will reach puberty in the home with the husband, and is it permissible to breastfeed her so that she becomes a محرم (unmarriageable kin), or is there another solution?

The prohibited form of adoption is attributing a child to other than his father, and this does not establish mahramiyyah (prohibition of marriage) or inheritance. As for taking a girl into one's care and raising her without attributing her to other than her father, this is permissible and one of the most complete forms of sponsoring an orphan. Breastfeeding this girl does not create tahrim (prohibition of marriage) if she is over two years old, because the condition for breastfeeding to create tahrim is that it occurs within the first two years of life. If the girl reaches puberty, she is to be treated as a non-mahram (stranger), so it is not permissible for your husband to be alone with her or to see her without hijab.