Is the brother's wife considered a kinship tie, and is it permissible to cut ties with her due to backbiting and causing problems, knowing that the questioner's relationship with her children is good and his brother is content?

The brother's wife is a member of the family, and it is recommended to maintain ties with her, treat her kindly, and do good to her, unless she persistently commits a forbidden act. Backbiting is a major sin, and she should be advised to abandon this characteristic. Shunning those who commit sins is for the purpose of reforming their condition. If shunning leads to the reformation of the sinner's condition, it is good and may even be obligatory. However, if interaction leads to that, it is preferable to shunning. One who openly commits major sins should not be shunned unless one is unable to deter them, for the ultimate goal is reformation.