What is my responsibility when I know that a person is sinning within the church?
The church that I attend has been hit with past and present members being charged by the law for pedophilia: one member has received a seven-year jail sentence; there is a woman who once worked in the church's children home for orphans and I know she has been having sex with boys, she escaped the courts by having a good legal team.
The church has done nothing, as she is the sister-in-law to one of its elders, I believe the Lord has taken her husband's mind as he, at age fifty, doesn’t know anything and now lives in a mental hospital.
Should I openly expose her and bring it to the attention of the Church or let her go on sinning?
The question if you know a person is sinning what is your responsibility?
This is one of those questions that I could take in so many directions and still will feel that I didn't fully answer all the issues raised.
When a person knows that sin exists, God is very clear as to what must be done: "And have no fellowship with the unfruitful works of darkness, but rather expose them. For it is shameful even to speak of those things which are done by them in secret. But all things that are exposed are made manifest by the light, for whatever makes manifest is light" (Ephesians 5:11-13). In fact, to know and say nothing jeopardizes your own salvation. "Son of man, I have made you a watchman for the house of Israel; therefore hear a word from My mouth, and give them warning from Me: When I say to the wicked, 'You shall surely die,' and you give him no warning, nor speak to warn the wicked from his wicked way, to save his life, that same wicked man shall die in his iniquity; but his blood I will require at your hand. Yet, if you warn the wicked, and he does not turn from his wickedness, nor from his wicked way, he shall die in his iniquity; but you have delivered your soul. Again, when a righteous man turns from his righteousness and commits iniquity, and I lay a stumbling block before him, he shall die; because you did not give him warning, he shall die in his sin, and his righteousness which he has done shall not be remembered; but his blood I will require at your hand. Nevertheless if you warn the righteous man that the righteous should not sin, and he does not sin, he shall surely live because he took warning; also you will have delivered your soul" (Ezekiel 3:17-21). How a person responds to a warning is between them and God, but whether you deliver the warning or not is between you and God.
I don't know the church you are referring to, nor the parties involved, so what I am going to lay out is a general procedure. I'm qualifying my remarks because the way your note is written makes me suspect there is much more going on that just what you are asking. Those hidden things might alter some of what I say and what you need to do.
When there is a problem between two Christians, the Lord laid out these steps: "Moreover if your brother sins against you, go and tell him his fault between you and him alone. If he hears you, you have gained your brother. But if he will not hear, take with you one or two more, that 'by the mouth of two or three witnesses every word may be established. And if he refuses to hear them, tell it to the church. But if he refuses even to hear the church, let him be to you like a heathen and a tax collector" (Matthew 18:15-17). If I could summarize the command, the Lord tells us to minimize the number of people involved to solve a problem. If I have a dispute with my brother and I can resolve it with him directly, then the matter is resolved and no one else needs to know about it. It is the concept behind: "Brethren, if anyone among you wanders from the truth, and someone turns him back, let him know that he who turns a sinner from the error of his way will save a soul from death and cover a multitude of sins" (James 5:19-20). Covering sins doesn't mean that sins are swept under the rug; rather, sins are dealt with and forgiven and thus forgotten (covered). Too many times I hear of sins supposedly forgiven but others insist on telling other people -- gossiping, ruining a person's reputation to no purpose, and preventing them from recovering from the disaster caused by their sins.
If, by "exposing the woman" you mean "tell everyone you meet what you suspect," then that is gossiping. As a Christian, your goal is to eradicate sin. It is not to grind a person's reputation into the dirt. Take the matter to those who can deal with the problem.
In the case you have outlined, it appears that you have not been personally harmed, but are aware of sins that are harming a third party. You state that you know these things are going on. If so, document your evidence as if you were going to court. Put yourself in the other person's place. If you were innocent, would you want accusations made based on rumors? Would you not want the facts presented? "And just as you want men to do to you, you also do to them likewise" (Luke 6:31). Paul told Timothy that when dealing with leaders in the church that evidence should be carefully collected. "Do not receive an accusation against an elder except from two or three witnesses" (I Timothy 5:19). An elder's position and responsibility make him an easy target for all sorts of accusations and rumors. Thus, care must be taken not to consider a report unless it is provable. I believe that such should be done with all members where it is humanly possible.
A witness is not just someone who has seen a sin take place. A witness can be the evidence left at the scene of a crime. It can be statements overheard. The two or three means there need to be multiple independent pieces of evidence that the sin exists.
Paul scolded the Corinthians for not handling problems that arise in the church. Any group will have problems. Christians are not immune from sin or the presence of sin. "Dare any of you, having a matter against another, go to law before the unrighteous, and not before the saints? Do you not know that the saints will judge the world? And if the world will be judged by you, are you unworthy to judge the smallest matters? Do you not know that we shall judge angels? How much more, things that pertain to this life? If then you have judgments concerning things pertaining to this life, do you appoint those who are least esteemed by the church to judge? I say this to your shame. Is it so, that there is not a wise man among you, not even one, who will be able to judge between his brethren? But brother goes to law against brother, and that before unbelievers!" (I Corinthians 6:1-6). Paul, here, is talking about disputes between brethren, but the principle remains the same. The court of brethren is independent of civil courts and ought to be superior to civil courts because our laws are vastly superior to the man-made laws of human governments.
You mentioned that one sister was found guilty of pedophilia and is serving a seven-year sentence. For the church, the matter is to determine if the government courts were accurate in their findings, which means reading over the court records and the court's conclusions. I would guess that it will be found to be an accurate case. Membership should be immediately withdrawn and efforts made to have the sister repent of her sins (I Corinthians 5:1-13). If she truly repents, then she must be welcomed back in (II Corinthians 7:8-11). It cannot be a sorrow of being caught, but a change that leads to a change in life. I know that it is currently popular to state that once a person is guilty of being a sexual predator that they will never change and must be forever marked. God tells us only one sin is unforgivable and that is blaspheming Holy Spirit (Matthew 12:31). Therefore, even a pedophile can be restored. Our only difficulty is making sure the change really takes place. We don't want to be found guilty before God of not forgiving a brother or sister who has truly repented (II Corinthians 2:1-11).
In the second case, you claim that the woman got off because of good lawyers. I won't be so quick to jump to such a conclusion. I would want to see the court documents. Was it because of a technicality? Was it a lack of solid evidence? Or was there sufficient counter-evidence to say that there was doubt in the court's mind as to her guilt? If you believe she is guilty, again, gather your facts and present them to the elders and the church. If the elder is doing his duty, family ties will not make any difference to him. Besides, churches have multiple elders to hinder this very thing. The woman is not the sister-in-law to all the elders. If you are factual and present a solid case, then you will find that people will not ignore the issue.
Sometimes people assume nothing is happening because they are not being involved personally. Take your concerns to the elders. You might learn that they are actively working toward a solution, but they are trying to spare the woman's reputation if possible. However, if things are already public because of the court trials, the elders should make their efforts known to the church. In this case, more harm comes from a sin being widely known and the efforts to restore the sinner is being quietly pursued.
Finally, it is very dangerous to assume that God is punishing someone because they are in a mental hospital. To make such a judgment is to claim that you know what is going on in the spiritual realm. "Even so no one knows the things of God except the Spirit of God" (I Corinthians 2:11). Until the Judgment Day, when all things are revealed, we must acknowledge that we don't know what God is or is not doing behind the scenes. Stick to what God has told us. He has told us what we must do and not do. He has given us a standard for behavior. Give righteous judgment based on fact, not what it appears to be to you. "Do not judge according to appearance, but judge with righteous judgment" (John 7:24).