The problem is that you want the strength to keep holding out hope...and that isn't a position of strength, but one of pining away and weakness. Want to be someone that she'll want to be with? Then your strength has to be someone that is complete and happy on his own (and with his kids); someone that will be fine without her, even if you prefer to be with her. But it's early yet, so it's understandable that you feel so weak. Life does go on. Your world didn't end because she's chosen another path.
It's almost certain that her feelings for this other guy have gotten in the way. I think the LRT is the right way to go. But that doesn't just mean that you give her space...it also means that you use this time constructively. There was a time before she ever entered your life. You weren't moping around waiting for her to come into your life then. Do things for yourself and with your kids. Rediscover the joy in your life. Work to be a complete man. If she doesn't want what you have to offer, it isn't your fault or your problem...it's hers. I know how crushing it is right now, but the sooner you start making a list of personal goals you want to accomplish, the quicker you can detach and realize that you can't control what she chooses to do.
You cannot be lonely if you like the person you're alone with. Dr. Wayne Dyer