i have 3 kids, also. a 3 1/2 yr old who doesn't nap anymore (boo!), a 2 yr old and an 8 month old. i JUST started exercising. i wasn't unhappy with my body after any of my girls, but i knew for health reasons I should be active. so, i'm starting off slow.
i bought for $10 a pilates dvd at target. it has five 10 min sessions that you can create a custom 'workout' with. i say workout with quotations because i know i'm not really pushing myself to do this dvd....i barely break a sweat. it came with a resistance band, too, which was nice. :) i can break out and do stretches with the band any time of the day without having to pop in the video...i've been doing that more often than the actual dvd.
anyway, i've been doing the exercises while the girls are up, playing around me. they think it's fun to try and do the moves, too. it's nice cuz i can pause the dvd if somebody needs something, but i'm still able to go back to it (without having to miss anything, like if i was in an actual class). this is all i can do for now. money is tight, so i can't afford to do any classes or gyms. i do feel really nice afterward and i've already started feeling better with my arthritis because of the flexibility sessions. i am nursing my youngest and she (STILL) wakes up at night, so i feel you on the being tired thing! the pilates helps me wake up if i do it in the morning while the girls are having breakfast....or if i do it in the afternoon, i get a nice burst of energy to start cleaning or start dinner. :)
my husband gets home late often, and when he gets here we just want to do family stuff and i'm too pooped to get up and exercise.
good luck with however you go about it. i know it's tough to work it in. just don't get down on yourself if you miss a day. i found that if i did, then i was less likely to enjoy it. :)