What is the Sharia ruling regarding a wife who confessed to committing an abomination with her husband's brother-in-law, and is it permissible for the husband to kill his brother-in-law? And is there a Sharia ruling for the wife despite his forgiveness of her? And what about the impact of this on the daughters' reputation in the future?

Disregarding the mixing of relatives leads to corruption, and prevention lies in adhering to Allah's law. If the wife has committed an unlawful act with a non-mahram man, she must repent. If she repents and improves her conduct, there is no harm in keeping her and treating her well, while striving to protect her and teach her religion, and preventing non-mahram men from entering upon her. As for the sister's husband, he must be advised to repent, and it is not permissible to kill him, for the establishment of punishment is the prerogative of the ruler.