How can one forget what happened and endure life after the wife finds out about her husband's relationships with other women, especially since he did not keep his promise not to repeat it, coupled with the fear of the family falling apart?

It is not permissible for a wife to pursue and uncover her husband's secrets, nor is it permissible for a husband to establish a relationship with a woman who is a stranger to him by any means whatsoever, due to the corruption that results from such actions. It is advised to accept the husband's apology and his commitment not to repeat the act, for the sake of the children.