With a couple of whiskys under my belt I would put the classic "Be a wife or outta my life" type of discussion on the table. If she's a wife she's going to act like one. If she's not then politely ask her to leave and set a date, approx. 30 days in the future. If it helps then find her a cheap apartment. She can't keep her toyboy since she'll be working to support herself.
I forgot to mention that I've seen this type of activity in Sweden before. It's not pretty. Women who marry for a meal ticket and then fob the kids off or get another man to have more kids with. Ugh.
Like I said earlier. Focus on the kids. You and them can be a happy family and perhaps you can find another woman to help. Either you have a marriage or you don't. She needs to think about that. Tell her that you know. Don't tell her what you know, just that you know she's not been your wife for some time and that someone else can support her in the manner in which she's used to. Perhaps her cellphone buddy?
Have a nice weekend! I'm off for more whiskey and a little grilled steak (which reminds me I need to get it off the grill!!)
