Is it considered a sin to treat the husband's sister in kind, after she had treated her like a sister and entrusted her husband to her care, but the husband's sister continues to be at odds and quarrelsome for no reason?

It is permissible to treat an oppressive sister in kind, but forgiveness is better, for "the recompense of an evil deed is an evil deed like it, but whoever pardons and makes reconciliation - his reward is due from Allah." Forgiveness is closer to piety and transforms an enemy into a friend, and increases one's honor. Patience in the face of harm and forgiveness are commendable acts for which one is rewarded, for the forgiver sleeps peacefully while the victor churns things over. It is advised to continue treating her kindly, to entrust her affair to Allah, and to avoid interacting with her except out of necessity.