Is it obligatory for the questioner to allocate a share of the house's rent to the adopted girl who is mentally ill, knowing that the mother registered the house equally between them with the aim of obliging the questioner to care for her, and that the girl is fully cared for in a special center?

Adoption is forbidden. It is permissible to care for foundlings and be good to them, but there is no obligation to secure their future. What the mother gave to the girl during her lifetime is her property. If she is a minor and suffers from a mental illness, the Muslim judge or his representative takes charge of her money. Her money can be spent on her education, such as memorizing the Qur’an, adhkar (remembrances), and hadith, and teaching her language and arithmetic. If there is no reason to spend from her money, the money should be preserved and grown for her. It must be noted that she is a stranger (non-mahram) to your sons. It is advisable to consult the Sharia courts.