How can one reconcile the father's desires for his wife to serve him and his sister with the wife's desire to visit her family, given the mistreatment of the wife by the father and sister, and what is the way to solve this existing problem?

The wife has the right to independent housing, and she is not obligated to serve the husband's parents. Her volunteering to do so is a benevolent act for which she is commended. The husband must honor his parents while also being kind to his wife. If the wife agrees to serve his family and he is able to reconcile between them, then that is good. Otherwise, the wife remains in her own dwelling, and the husband must honor and be good to his parents.