There have been a lot of suggestions surrounding cutting out internet, putting up filters, getting him accountability, etc. The problem with this is that, even for the person who is addicted to inappropriate contentography and wants to stop looking at it, they have an addiction, meaning that they can't control it. An addiction isn't just a moral defect, it is something that literally rewires the brain. You indulge in the addiction, it triggers a release of chemicals from the pleasure centers of the brain, and your body takes that positive experience and says, "We need to do this again."
So when you are dealing with someone with an addiction who is not all that interested in stopping, cancelling internet or putting up filters is kind of like you babysitting your husband, and no one wants to have a parent-child relationship with their spouse. Along with that, it's like those cartoons where holes are popping up in a boat and the character is trying to plug them all to keep it from sinking. You can put a stopper on the internet, your computers, your phones, etc. but if he really wants it he is going to find a way to get it.
So with that said, the solution here is that he needs to make the decision for himself that he is going to make changes. I know your concern is about the safety of your children, and I will say that while his inappropriate content addiction and the nature of his searches don't necessarily mean that he is a danger to them, it would be naive to think that it isn't a possibility. If your husband has been a victim of sexual abuse himself in the past, that would make it even more of a concern in my mind.
But regardless of whether or not he is a danger to your kids, he is seriously hurting you with his actions and that is enough to demand change. Any advice you have been given that says you should continue to endure this for the sake of your children or not splitting up your family is downright horrible advice. You are not required to endure this type of emotional trauma by God or anyone else.
With that said, I think it is most helpful if you are clear and straight-forward but also kind in the way you speak to him. It is up to you if you want to wait until you have caught him again, or if you have caught him recently that is enough to have the conversation. My suggestion is to ask him to leave, and say that the only way he is allowed back in is if he starts making changes. There are 12-step support groups all over the country like Sexaholics Anonymous and Sex Addicts Anonymous that cost nothing to be a part of. If he can go to counseling, that would be ideal as well. But at this point, after 10 years, the only way that crap is going to get real for him is if he starts to sense that his addiction is causing him to lose what is most important to him. There is no guarantee that he is going to change, obviously, and you have to be prepared to divorce if that is the case, but this is the only real way you have at this point to ruffle his feathers enough to make him do something about his behavior. I know men who were engaged in extramarital sexual behavior due to their addictions, and one day crap hit the fan, so to speak, and they were kicked out of their house. They saw that they were going to lose their wife and children, and they started doing everything they could to reconcile and get on a better path, and for most of them that is exactly what happened. Many of them have been clean from their addictive behavior for several years now. That is what I hope would happen for him, for your marriage, and for your family.
This might seem a little bit extreme, but marital infidelity is extreme. You do not need to endure this kind of pain in your marriage just because your husband doesn't want to work at it. And all of this is biblical advice. I would not suggest anything that I thought was unbiblical. I will be praying that God would give you the wisdom and strength to do what is necessary for the sake of your own emotional well-being and your marriage, and that he would have a change of heart and find freedom from his addiction as well.
