I think I finally figured something out, tried it this morning and it worked beautifully. It seems so obvious I can't believe I didn't think of it sooner! I don't have a treadmill, but I do have a cul de sac (as an aside, this word has always sounded faintly pornographic) right outside my door. I go out there at 6:00 am and do laps up and down the street, around and around. The kids are usually asleep until at least 7:00, but they're old enough now that if they do wake up they can get themselves some breakfast or help Gavin if he wakes up. And if they do need me urgently, I'm just outside on the street.
Here's my little workout spot:
I really hope this will work! Week 2 of Learn to Run 10k starts on Sunday.