His wife became Muslim then she changed and got to know a non-Muslim man, and she wants a divorce
Question

                I have a question regarding a divorce. My wife who is a revert asked me for a divorce 8 weeks ago. She has moved out of the house for a month now and havent talked to her for 2 weeks. I recently found out that she is talking to a non-muslim and is practicing signing her new name with his last name. I have not given her Khula as I tried everything to make her get a divorce according to Sharia. What should I do now when she has gone so far away from Islam? Should I still encourage her to do her Idaa? She is also planning on traveling without a Mahram and I am not sure if she is going to go see the new guy or going with her female friends. She used to be really into Islam but now she has gone astray and I feel like Shaytan has got hold of her.
Praise be to Allah.
      Firstly: 

    It should be understood that when a woman asks her husband 
    for a divorce, that does not always mean that she is serious about ending 
    her marriage to him or that she no longer cares for him; rather it may be a 
    moment of sudden anger in which she says something but does not really mean 
    it. Or she may be going through difficult circumstances that made her say 
    this. Hence you should not be hasty in this matter and you should try to 
    remedy the problem in a suitable and wise manner. 

    But if the matter was as you describe, that she has started 
    to get to know another man and is in contact with him, then this is serious 
    and the cause of this calamity, especially if that person is a kaafir whom 
    it is not permissible for her to marry even if she did not have a husband, 
    so how about if she is the wife of a Muslim? 

    Secondly: 

    In the light of what we understand from your question, which 
    is that before this the woman was sincere in her Islam, what we advise you 
    to do is to be patient for a while regarding the matter of divorce, so long 
    as you do not allow her to continue being in touch with that man who she has 
    got to know or with any other men. Rather you should try using gentleness 
    and kindness with her, and try to draw close and to be friendly towards her, 
    not only with the aim of keeping her as your wife, but for the sake of 
    something that is more important than that, namely keeping her as a Muslim, 
    because there is the fear that she may turn back on her heels and apostatise 
    from her religion if you allow her to have her own way and leave her to do 
    whatever she wants. 

    So try by all possible means to prevent her from travelling 
    as she wants. Try to speak kindly to her and remind her of the life that you 
    have together, and of the blessing of Islam that Allah has bestowed upon 
    her; and remind her that it is not permissible for her to marry a non-Muslim 
    man if you do divorce her, so how could she form a relationship with him 
    when she is still married to you? 

    If you can perform ruqya for her yourself, or you can find 
    someone who can do that, because of what you think of the possibility that 
    she has been affected by the Shaytaan, then do that, because perhaps it is 
    as you think; then perhaps Allah will heal her from this calamity. 

    When you have done all that you are able to and have tried by 
    all available means to set her straight, then if she responds and you see 
    that she wants to repent from what she has done and adhere to her religion, 
    then keep her as your wife and strive to teach her about the religion and 
    help her to adhere to its rulings. 

    But if you find that she is persisting in what she is doing 
    and exhortation has not worked with her and you cannot find any way to bring 
    her back to her house and make her obedient to you, then divorce her and 
    leave her to what she has chosen for herself. If you want to keep her and 
    refuse to give her a divorce (talaaq), and she resorts to khula‘, there is 
    nothing wrong with what you have done, because she is wronging herself and 
    her husband and is being defiantly disobedient; in fact she is seeking that 
    which is damaging to her religious and worldly interests. 

    And Allah knows best.