For now you have made your move. You don't need to say anything new. You may need to repeat yourself a few times. Now you wait and see what he does. This could push them closer together or tear them apart. It depends on how committed he is to the affair. Some men just wanted a little extra on the side. Some have really fallen in love with the OW. Time will tell. He can go one way first, and then change. Stick to your guns.
As for what to tell the children. I believe in the truth, not necessarily the full adult truth, but some version that is truthful. Perhaps, "Daddy needs to think about what he wants. When he makes up his mind he will let us know." Others with young kids will have better ideas. My kids were old enough to just tell the whole truth.
As for telling your family, you might tread lightly there. Sometimes the family has a harder time forgiving later on than the spouse does. Maybe just " we're having trouble", if you can get away with that.
