How can one atone for shortcomings in caring for a deceased mother, given that the wife refused her permanent residency and her residency was equally distributed among the children, and one of the siblings refused to host her?

You are not required to keep your mother alone without your siblings, as caring for her is obligatory for all children according to their capacity. As long as you have done your best in showing righteousness to your mother, you are not sinful. If there was any shortcoming, it can be remedied by supplicating for her, seeking forgiveness for her, giving charity on her behalf, maintaining kinship ties through her, and honoring her friends.