How can a mother be persuaded to live with her married daughter, and what is the ruling on compelling her to do so for her own benefit, given that she rejects the idea without justification despite the husband's approval?

You must be dutiful to your mother, care for her, and protect her, which includes not leaving her alone. It is best that you convince her to live with you. If she refuses, then hire a maid for her if she needs one and is unable to afford it. It is not permissible to force her to do something she does not want. To convince her, ask her to visit you for a few days, honor her, and show joy at her presence. Perhaps she will feel comfortable and be convinced to stay with you. You must always commit to visiting her, contacting her, and asking about her.