This is the very reason that I do not want to have kids. I don't think the marriage is the issue--you've had kids and now you have to make sacrifices (rightly so).
That said, I'm a 34 year old woman. I live in Central Florida, where the temps rise to mid-90's in the middle of the day and humidity is also ridiculously high. I have run many times in the middle of the day--including some track work. It is not pleasurable and may even be harmful (which is why I'm trying to stop), but I'm not dead yet, so it can be done. I was extremely comitted and that is when I could fit in my run, so that is when I ran.
I still think talking to her would lead to a solution. You have to make it clear that this is very important to you (and possibly give up other hobbies or whatever that aren't as important so that you can spend time with her and the kids when you are not running). I like the idea of your doing other things with/for the kids when you are not running, such as preparing lunch at night or laying out the kids' clothes.
Sweet talk is also very effective. The sexual comments on here are ridiculous--women don't want sex when they are pissed about something else. The sex comes after you've found a solution via sweet talk and compromise.
Good luck.