Is it permissible to kill a wife who practices sorcery and harms others, including her husband and children? And is this considered a solution to confront her criminal acts?

If what is attributed to your brother's wife is proven, then her actions are among the major abominations (such as frequenting sorcerers, going to church, performing magic, and spreading rumors about her husband's impotence). These actions do not permit the killing of a believing soul, as that is one of the greatest major sins. As for her enabling the sorcerer to commit adultery with her, if that is proven by four witnesses or by her confession, then she deserves to be stoned. However, the implementation of this Hadd (punishment) is the prerogative of the ruler or his deputy. If this grave sin is not proven, then whoever accuses her deserves eighty lashes. The source of the information should be verified, especially if it comes from an exorcist who might be a charlatan.