How am I to deal with having a disbelieving spouse? We were married in the temple 13 years ago and we have 4 kids. She used to be strong and now she just doesn’t believe any more after years of going behind my back on anti-Mormon sites. She has chosen to hear “their” perspective on issues rather than mine. I believe marriage is a covenant I should not let go of. But where is joy found? How do I even smile when the only thing I want (a faithful in the faith companion) is not there?
You handle a disbelieving spouse the same way you handle any challenge you might face in life. Namely, you spend a lot of time praying. You spend a lot of time studying the scriptures. You spend a lot of time pondering on how the Lord would have you respond to the various obstacles presented.
You are not the first person to find your spouse to be a problem when you were hoping to have a helpmeet. Think of it as a chance for you to develop greater Christ-like attributes. Learn to love her for who she is, not for what you want her to be. Just like Christ loves all of us. Your joy should come from your efforts to be more Christ-like.
Having a disbelieving spouse does pose additional challenges for your relationship and for raising your kids in a gospel centered home. Do not be afraid to ask for help from spiritual experts (like your bishop) and relationship experts (like therapists) if you find yourself stuck trying to solve some pretty difficult issues that might come up from time to time.
Of course your efforts should also include trying to help your wife return to the gospel. In this you should remember how much God allows us to exercise agency, and you should take the scripture’s injunction to act “by persuasion, by long-suffering, by gentleness and meekness, and by love unfeigned; By kindness, and pure knowledge.” In your attempts, make sure you do not violate her agency. You state that your wife looked at anti-Mormon material. I have not seen any anti material that can’t be countered. Yet something within her caused her to seek that stuff out. Whatever that is, it is the root cause and all the counter material in the world will not help until it is discovered and addressed. So it seems to me that your first step is to listen to your wife’s heart and see what is really going on there.