All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
What you are experiencing now is the best example for the danger of going against Allaah's Commandments and the orders of the Prophet . Allaah Says (what means): {And obey Allaah and the Messenger (Muhammad and beware…}[Quran 5:92]. Ash-Shawkaani interpreted the verse to mean: "Beware of going against Allaah's Orders and those of the Prophet ."
On the other hand, residing in a non-Muslim country is a widespread evil and it has bad effects on a person, his wife and children. Indeed, the Prophet said: "Whoever keeps company with a non-Muslim and resides with him, is like him." [Abu Daawood] The Prophet further said: "I am free from any Muslim who resides among the polytheists (non-Muslims)." [Abu Daawood]
Besides, it is forbidden for a woman to work in a place mixed with men as that leads to adultery and fornication.
Hence, our advice to you is as follows:
1) To repent to Allaah for migrating with your family to a non-Muslim country, if it was not due to necessity, while not being able to fulfil your religious obligations towards your family, and allowing your wife to work in a mixed place, because you are in a way a cause to such a sin and shame.
2) You have to remind your wife of Allaah and His severe Punishment. You may seek, in doing so, the help of whoever knows her and could influence her, and you should make use of Islamic appropriate tapes and lectures. It might be that she would repent and return to Allaah.
3) If she sincerely repents and it is evident that she has done so, then we do not see any religious harm in keeping her as a wife. However, if she committed adultery, she has to observe a waiting period, exactly the same as that of a divorced woman, and it is not permissible for you to have sexual intercourse with her until the end of her waiting period from the fornicator.
4) If she did not repent or you are sure that she has not sincerely repented, then you are obliged to divorce her because there is no good in living with an adulterous wife. Indeed, many scholars are of the view that it is forbidden to keep an adulterous wife because Allaah only permitted us to marry chaste women. Allaah Says (what means): {(Lawful to you in marriage) are chaste women from the believers.}[Quran 5:5] Allaah also Says (what means): {The adulterer marries not but an adulteress or a Mushrikah (a female polytheist, pagan or idolatress).}[Quran 24:3]
In case divorce takes place, then the right of fostering the children is yours (the father). Since the mother is a dissolute person, that makes her lose her right in fostering her children. One Hanbali scholar said: "A slave, or a dissolute person, or a non-Muslim has no right in the fostering of children, and a woman who is remarried with another man than the relatives of the children loses the right of fostering them as well, as soon as the marriage contract is signed."
So you have to look for a righteous wife who would help you in your religious matters and in bringing up your children and you should forget this bad experience. Furthermore, you should not feel sorry for an adulterous wife who did not fulfil the Rights of her Lord and the rights of her husband. Instead, you should be keen in bringing up your children according Islamic principles and learn a lesson from what happened to you, because a believer should not be stung twice from the same hole [undergo the same problems twice], as per the saying of the Prophet .
Allaah Knows best.