What are the mutual rights and duties between a wife and her husband's family, and how should a wife deal with her husband's family members who do not reciprocate affection, do not inquire about her daughter, and differentiate in their treatment between her and their other grandchildren?

It is recommended to be patient, forgiving, and to repel evil with good, for this transforms enmity into warm kinship. Part of loyalty to the husband is loyalty to his family. Problems are dealt with wisdom and kindness through pleasant words, smiles, gifts, and fulfilling their needs. If their treatment does not change after effort has been expended, then you have fulfilled your duty, and your reward is with Allah. You may then limit your interaction with them to the bare minimum during occasions.