I hope that God is blessing you in what you do.
I am writing you on the matter of infidelity. I am aware of a man who has been unfaithful to his significant other. This is not the first time that he has been so. The previous time which he was unfaithful he admitted to her. However, the second time she does not know of it. I know that he has asked for forgiveness from God, but he does not admit or tell her of his other infidelity. Even if she has said she would not want to know, does that mean God's forgiveness will not apply to him?
When someone says "significant other," it usually means they are not married. If that is the case, then there are no obligations between the two people. Both are committing fornication when they have sex together and they are committing fornication when they have sex with others. "Do you not know that the unrighteous will not inherit the kingdom of God? Do not be deceived. Neither fornicators, nor idolaters, nor adulterers, nor homosexuals, nor sodomites, nor thieves, nor covetous, nor drunkards, nor revilers, nor extortioners will inherit the kingdom of God" (I Corinthians 6:9-10). The idea that God would forgive one set of sins while the identical sin continues with a different person is contrary to what the Bible teaches. One has to repent of sin (reject the sin and turn away from it) to be forgiven (II Corinthians 7:10-11).
Forgiveness of an individual's sins does not depend on the forgiveness of people he harmed by his sins. It is the people who are harmed by a sinner who is told to forgive the sinner when he repents. "Take heed to yourselves. If your brother sins against you, rebuke him; and if he repents, forgive him" (Luke 17:3). When a person is unwilling to forgive someone who has harmed them, their own salvation is in jeopardy. "For if you forgive men their trespasses, your heavenly Father will also forgive you. But if you do not forgive men their trespasses, neither will your Father forgive your trespasses" (Matthew 6:14-15).
A sinner must repent of his sins and confess his sins to God. "If we confess our sins, He is faithful and just to forgive us our sins and to cleanse us from all unrighteousness" (I John 1:9). The sinner is also encouraged to talk to fellow Christians about his sins. "Confess your trespasses to one another, and pray for one another, that you may be healed. The effective, fervent prayer of a righteous man avails much" (James 5:16). But there is no requirement that every victim of our sins must be contacted. On a practical level, it can't be done. Sometimes the person is not reachable or even has died. Sometimes the person doesn't want to talk to us. Sometimes we don't even realize we hurt someone.
People who repent often do talk to the people whom they hurt. They do so in a small attempt to repair the damage that they caused. But there are times when it is best not to say something. For example, suppose you said a really degrading thing about one of your friends, but your friend wasn't there and does know what you said. Bringing it up will hurt your friend and you've repented of hurting him. Thus, in such cases, it is best to bury the sin that has been repented of and forgiven by God. "He who conceals a transgression seeks love, but he who repeats a matter separates intimate friends" (Proverbs 17:9).
It is that last verse that you are contemplating violating. Your friend has repented of his sin. He has sought God's forgiveness. Yet, you haven't forgiven him because you are thinking of keeping his past sins present. His wife doesn't want her view of her husband ruined, so don't do it over a matter that over and done with.
My answer would be different if you knew he is continuing to sleep around. Then she needs to be warned that she is being deceived.
If she finds out later on her own and chooses to ask her husband, he must tell her the truth, but he also can tell her that he regretted his deed, cut his ties, changed his life, and is not the man he used to be. She would know this from his behavior and it will make it easier for her to accept that he repented and be able to forgive him. However, it is not your place to interfere.