Is it permissible for my wife to ask for a divorce or to move to another residence far from my family's home, knowing my sick mother's need for me and my desire to honor her, and what is the best course of action in this difficult situation?

You have provided your wife with independent accommodation, which is her right. Therefore, you are not obligated to move to another place, and she does not have the right to separate you from your sick mother. Honoring your mother is an absolute obligation, and obedience to the husband is obligatory for the wife. Requesting a divorce without a legitimate Islamic reason is forbidden for her, and she will not even smell Paradise. If she continues to refuse to live near your family, she is disobedient (nashiz). Beware of neglecting the honoring of your mother and staying close to her, for "Indeed, among your wives and your children are enemies to you."