What should a wife do who discovered her devout husband's infidelity, and he confessed to it and promised to repent, but she cannot forgive him or forget the matter and feels doubt and breakdown, and the husband refused divorce for the sake of the children?

If the husband has a relationship with a foreign woman, he has committed a reprehensible act from which he must immediately repent and sever this sinful relationship. If he repents, then praise be to Allah. Otherwise, it is recommended to separate from him, and one can resort to the Sharia court if he refuses divorce.