I don't want to be demotivating, Sophie, but if you need sleep, you need sleep. I think it's more important than exercise. Get the shut eye in first and foremost!
I'm a full time carer at the moment and can be up at all kinds of strange hours, sometimes the whole night. Like you, I find it hard to keep body and soul together sometimes. Sleep is really difficult for me because I tend to drink a lot of water and tea when I'm up, then when I finally crash and try to get a nap, I'll be running to the loo on the hour. Don't think I've managed more than 2 consecutive, uninterrupted hours of sleep in almost a year. I try to embrace the poncily titled 'power nap'!

If I can grab 15 minutes on the sofa and drift off, I do. Your whole perception of reality and your outlook can be so dramatically altered by sleep deprivation. It's scary how low you can get when exhausted, so I'd always put sleep ahead of the gym.
I don't do any formal exercise these days, but I've used dvds in the past and I found them rather fun... certainly better than having to head to the gym. Dan Karaty's hotpants workout is a good one. I tried Davina's but it drove me nuts. The Ministry of Sound one was okay-ish. My favourite exercise was always dancing, though. Just put the earphones in and scare the postman with your boogilicious stylings!
For food... it's a minefield. The more tired you get, the more attractive the nibbles become. Another good excuse for regular napping! All I can say is just try to keep ahead of your cravings and weak times and have plenty of healthy snacks ready to be demolished.
Hope things start to fall into place for you.