Lucky for you, there's a member on this forum that has fairly severe sleep issues that could probably offer you some good advice...
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...man, who was it though? SIGH, if only I could remember who it was. Oh yeah, that's right!!! It's...ME!!!
Anyway, I've been dealing with sleep issues for quite a long time now and I know how bad it can suck not getting any rest. When my body begins to feel sleep deprived, it becomes an extremely huge waste of time if I workout. Nothing beneficial comes out of it.
First of all, I'm way too tired to put forth an appropriate amount of physical effort in order to burn any calories, so any effort I would put in wouldn't be worth my time. Secondly, my body is way too messed up for things to be working properly. My body is trying to heal itself, which is something that it normally does while I'm sleeping. So, if I'm not getting enough sleep, my body isn't healing itself enough. And, if I try to workout with a damaged body (so to speak), I'm only going to be making myself weaker and more damaged. So, working out while I'm exhausted is only going to do more harm than good. Last, but certainly not least, when my body feels sleep deprived, I become an emotionally unstable, cranky mess. I don't know if you've ever tried to workout while in an unstable mental condition, but it's not fun. It doesn't make you feel good, like workouts normally do; it makes you pissed off at the world and ignites the fuse which can lead to emotional eating. And, if I start eating emotionally after a workout, then I'm just taking 1 step forward, 2 steps back.
So, in my experience, working out after your body has suffered a true lack of sleep is just a huge negative.
My recommendation would be the same as Jericho's - worry about catching up on sleep first, before you focus on losing weight.