I was in a similar situation. Depressed wife at home, but no kids. Nothing seemed to work for her depression, then she decided cheating would make her life interesting. We divorced, it was nasty. I'm happier now then I ever was married. I'm thinking about marrying again, but their are some real risks.
You are doing the right thing by going through counseling - you owe that to yourself and your kids. But if she isn't putting in the effort, you also owe it to yourself and your kids to try to make a clean, gentle break. It's tougher because you have children. The two of you will need to be civil so that you can continue to parent the kids together. If you are successful, it could be better than continuing in a strained marriage, which isn't healthy for either of you, or the children. But it's a big if.
I live now with my girlfriend, and we've talked marriage. She has an 8 year old daughter, and she and her ex had a civil divorce. We all manage to act like adults, and as a result I think her daughter is better off. Her ex has remarried, and we are together. As a result, her daughter gets to experience life with two parent couples who have functional relationships. I like to think that it is better then her daughter experiencing life with one disfunctional couple. I may be wrong.