How can a wife be advised to play her role in caring for a sick mother, especially given previous discussions and the insufficient involvement of the brothers' wives?

The wife is not legally obligated to serve her husband's parents, nor is it incumbent upon her. However, it is not considered part of "living together in kindness" (al-mu'ashara bil-ma'roof) for the wife not to serve them, especially during illness. There is no objection to the husband requesting this of her or marrying her for this purpose. The wife should know that helping those in need, especially the husband's relatives, brings her reward and recompense.