Should the family of a wife who commits adultery be informed, given the husband's alleged inability, and what is the ruling on remaining silent about it?

You must advise her and instill in her fear of Allah's punishment for adulterers, and send her some fatwas. If she does not desist, inform her family members who are capable of preventing her from committing the sin, using allusions such as "Pay attention to so-and-so," without explicitly accusing her. This is unless four just witnesses clearly observe the crime. By doing so, you would have fulfilled your obligation, and you are not required to testify against her.