You know, a lot of people are going to come on and say "Throw her out!" or "Man up!" but I'm going to say something different.
This is up to you. You need to decide what you are and are not willing to put up with--and then live with your decision.
Look, we don't live in your skin or at your home. None of us have to look at your kids as you put them to bed at night. None of us even KNOW you! You do. You know you; you live in your skin, in your home, with your kids. You know what you can stand and what you can not.
So I would, at minimum, suggest that you ask her to stay with a friend or with her mom until you've had some time to yourself to think. I would possibly even suggest that you take a day or two off work so you can "get away" and really think about yourself, what you want, what you believe, who you want to be, how much you can endure, and what you want to be for your kids. THEN based on those answers, make a plan how you intend to act, and then ACT. Be deliberate and be in line with the Inner You. We can't decide that for you or even tell you what to do--only you can do that. Some people might be able to endure 6 months of waiting...some 6 weeks...some "decide now, leave or go". Really what we can tell you is that choosing to have an affair to address the issues in your marriage is definitely wrong. Choosing to continue an affair after you're found out is definitely wrong. Choosing to re-contact after claiming that you're trying to reconcile--even if it's in grief over the death of your father--is wrong. So we already know she has done wrong, and we already know she is willing to lie and place the blame "on her grief"
The question is this: do you choose to reconcile?
I can say that I won't be one encouraging you to just end your marriage. I can also remind you that unfaithfulness is the one moral, ethical reason given in almost every religious and legal system I know to divorce your spouse with a free conscience. You have the right--there is absolutely no doubt of that. The question is whether you are going to choose to exercise that right. I can also say on a personal level that if it were me, I'm pretty sure I wouldn't just "end the marriage" but I sure as heck would count that as an enormous
boundary crossing with consequences that matched that sort of devastation. I would probably go with, at minimum, a separation of X endurance, and whilst holding VERY little hope, give that X endurance to see exactly what deep and dire changes were made...with an attitude a little bit of "I'm not helping one bit but you have X months to prove to me it is or is not worth it to exercise my right. The end." Finally I can also say that I would be someone who encourages you to forgive her--not based on whether or not she apologizes or is remorseful, but based on the fact that holding onto the pain and grudge will hurt YOU. Whether you two reconcile is irrelevant--even if you divorce, at some point forgive her for yourself...and let her deal with her demons.