Go for it, I'll be glad to part of your support team. I'm a guy, so hopefully you can get some gals to help out also. Recommend your primary emphasis be on the "feel good" objective. If you can make this one happen, the weight one will come along on its own pace.
Your exercise sounds good, I'm more of a "walking" type due to a bad knee, so do my sprints on an Air-Dyne. Suggest doing the sprints 1 or two times a week. I'd stage these and do something like my baseball coach back in the olden days had us do in sprinting the straights, walking the curves on our track. This is the concept, doing it on a track is way too boring, but that's the idea, go like crazy, then let yourself rest up and go again, repeat till gassed.
I'm not a weight training type either, so can't provide any advice here. Suggest browsing for info, there is a serious group of weight training people here, but I think more in tune to the actual weight room experience than just body weight stuff. I suggest looking at Mark's intro materials, he has the four core exercises. If you can do planks, squats, sit ups, push ups, and pull ups (I know, that is 5), that will give you a good work out. Start easy, work up repetitions and possibly add some weight as needed. I'd also explore and find some flexibility exercises, as a runner you may need to especially pay attention to your legs. Back "in the day" I always had issues with my legs getting tight when I would be doing a lot of running to stay in shape for baseball and football.
Best way of handling the food is to get the whole family on board, that way you can cut out the junque you buy at the store so you don't have a lot of temptations in the house. Don't know how easy this might be with your kids, but maybe you can work to phase them in.
The Buck stops here. I am responsible for my past and my future. So for today: I choose to be happy. I will seek wisdom. I will be a servant to others. I will greet this day with a forgiving spirit.