Whatisis is helping you to think practically. Listen.
Let's look at at four things:
1. What's killing you right now? I mean what's REALLY killing you? The lack of respect. Your wife is screwing some other guy. In another time or another culture, I might join the lynch gang and string that bastard up with you. In understand. You feel like a doormat. You feel emasculated, you feel like you are your wife's bitch. You want you power pack. You feel like your boundaries are being stepped on. You want to feel like a man, like a freaking human being. By right, by natural law, by every fibre of your being, by your sacred honor the one thing you feel should happen is white-hot justice to be executed. A 180 for you would be, in some way, being more assertive, more clear, more powerful, more focused, more Alpha-Male. Is asking her to leave the house the ONLY way to achieve that? Is it? Are there other ways of declaring boundaries? Think hard.
2. What if she says, when you ask her to leave, "No, it's my house too."? What then? What's your ultimatum? You can't make her leave. The courts will proabably ask you to leave if you file for separation/divorce. Will you leave if she says no? And then what happens? You leave your own home, subsidize your wife's affair and punish your children. I'm not saying asking her to leave is a bad idea. If she wants to have an affair, she should leave your family and feel what's it like to lose something. It might do her some good. It might make her think. But -- I think asking her to leave means you are, to some extent, really prepared to follow through with somethign more drastic. Do you want to have custody of the kids? Do you want shared custody? Think hard about the worst case and figure out what you want. If you want shared custody then a nasty divorce is not necessary. If you want full custody, then you need to play your cards differently. Go talk to a lawyer. Get an accurate picture of what will happen given various "what-ifs". Be careful, divorce lawyers are often unscrupulous. They want to it to drag out long. They will promise the moon and deliver a shared custody arrangement that you could have done with a mediator.
3. Patience. Are you sure your DB efforts are not working? Sometimes they take 6, 9, 12, 18 or 24 months. Only you can figure out how much is enough. Have you tried the Last Resort Technique? Have you experiment with various forms of detachment/being assertive? Are you really GAL to such a degree that she's getting curious about you? Remember it takes at least 3-6 months of GAL to be noticable.
4. I think you need to get to the point where you are no longer afraid of losing her. Once you are there, whatever you do, even if it asking her to get an apartment, will be done without fear, without desperation, without, frustration. You are still DB, but able to take risks. Are you in that place yet?