Bro Smoker your situation is similar to a friend of mine that happened many years ago .
Main differences the lady was a PRC KTV WL who made a lot of money and wanted my friend to divorce his wife and followed her and live happily in China. When my friend suggested to break off because he has 2 small kids and the lady is getting very sticky , the lady like most women threatened to commit suicide. She viewed his wife as the 3rd party.
The final straw was she called up my friend's wife and told her about their affair . Of course all hell broke loose.
My suggestions are:
1. Start if you haven't to be generous and nice to your wife's family - parents, siblings etc. Also do the same for your family and siblings because these people will back you up later.
2. Continue to meet up your GF because if you stop abruptly she will hit you back hard. Go out as usual for drinks and chit- chat but no sex. Slow talk and be nice. If possible go out with group of friends where she can meet up with other guys. As time goes meet up less often , hopefully she gets bored and look for some the other guys.
3. Be nice and considerate to your wife. Spend more time with her and the kids.
4. Finally when the time is right tell your GF that it's best she look for a better guy. Someone she deserves and can really love her. Give her some money as parting gift if she agreed.
5. Be prepared if she disagrees , goes ballistic and threatens to tell your wife. That's when you have to divulge to your wife what has happened - skip the gory details of sex- and do so with teary apologetic eyes. Pre-empt her what will happen.
6. Once this blows over be prepared that your every move to be scrutinized by your wife like a home-detainee . My friend suffered 3 years on guard by the wife even though she forgave him.
Hope this helps..