I know there is no way I could continue to live with a man that had a GF. What message are you sending your kids about how a woman deserves to be treated? If they see you are treated this way and you allow it to continue, they will both see this as acceptable. So down the road your daughter will not stand up for herself in a relationship and your son will think it is OK to cheat on his significant other or wife. That's only my opinion. I know you're in a hard place. Prayers and hugs are sent your way.
I absolutely agree with Becky. Don't let your children think relationships outside of the marriage are acceptable. That's what they'll believe if you continue on.
If your husband knows that you know about the gfriend, what does that tell you about his respect for you, your wedding vows, your marriage, your family.