She is asking you to fight for her. You should avoid at all costs begging and crying. But you need to let her know what your values are. That you value your family. You want an intact family -- mother, father, children, grandchildren.
You should woo her. Dinner out. Touching. Look her in the eyes when you speak to her. Use her name. Thank her when she does things like make a nice meal, or clean the house. Notice what she does. Ask her to do stuff with you, like the Fargo activities.
And most of all, get her to agree to go to a Retrouvaille weekend with her. (that's pronounced retro-vie.) But you can just call it a marriage retreat weekend. And as I've told others, it's fine to travel to another place to do it. That is the fair chance that you need. A weekend away together focusing just on working on the marriage. If she doesn't do it, she's not giving you and the family a fair chance. http://www.retrouvaille.org., the schedule of weekends is on the web.
And let the kids tell her what they think. Your marriage is not just between the two of you. They have a stake in whether or not there is a mother in their house. You don't have to tell them what to say. They'll figure it out just fine.
Don't let her think you don't care. She wants you to demonstrate that you care. She wants to know that she is not taken for granted. He doesn't take her for granted. You and the kids can't either.