What is the ruling of the Sharia on my wife and me separating to live in a residence close to my father, who treats my wife with harshness, knowing that my father is in good health but cannot tolerate serving himself, and my wife will undertake his service?

It is the wife's right to live in an independent dwelling, and she is not obliged to live with her husband's relatives or to serve his family. Therefore, there is no objection to separating into a dwelling close to the father, while being keen to honor him and check on his affairs. Wisdom should be exercised to avoid the father's anger and discord.