Oh honey, I feel your pain. My son was just like that. I'm sure he must have slept at some point during his infancy, but I don't remember it! My husband and I were so tired that we were walking into walls...it was not fun.
I wasn't able to breastfeed, so my situation was a bit different from that of a nursing mom. Still, here's what I can offer:
• I found most sleep books worthless, but the one that did actually help somewhat was The No-Cry Sleep Solution. I mention this because you say that sleep training is not working with him. There's a school of thought that some babies release tension by crying and some increase tension (more here: http://www.askmoxie.org/2011/01/tension-increasers.html). Perhaps you have a tension increaser. If so, you'll need to adjust accordingly.
• My son did not like sleeping on his back AT ALL. He wouldn't stay asleep more than an hour in that position. We toughed it out because we weren't willing to risk putting him on his stomach. Once he could roll over on his own, he slept better. Not great, but better.
• As suggested below, the Miracle Blanket is worth a try. It didn't help us; our baby escaped it very quickly. But it can't hurt.
• Have you tried white noise? We didn't start using it until our son was in preschool and I wish we'd done it in his infancy. It made a big difference. Lavender-scented bath products and lavender oil helped ever so slightly at the baby stage.
• What are your sleeping arrangements? If he's in a crib, are you willing to consider putting the crib in your room temporarily, next to your bed? Or using a cosleeper that's next to the bed but separate? Some babies sleep better with someone near them in those first few months and you can gradually increase the distance from your bed and your room as they learn to fall asleep on their own.
• For naps: I've got nothing. The only way I ever got my son to nap was in the swing or driving in the car. Not ideal, but I was desperate. I would sometimes get a book on CD and just cruise around.
• I agree with Megan about the overtired/overstimulated problem. You probably know all that since this is not your first rodeo, but we found that even making eye contact with our baby at bedtime was enough to get him fired up and alert.
• Do whatever you have to do to get a modicum of sleep at this point. If your baby will take a bottle, could you pump enough milk that your spouse/significant other can be "on duty" every other night and let you get as much sleep as you can? Even if you need earplugs or have to sleep in the guest room, you absolutely have to get some rest, especially if you have two other kids to take care of.
• Take comfort in the knowledge that very often, kids like this are super-smart, curious and lively. This has certainly proved to be the case for us. Their little minds are already clicking away, they're interested in everything around them, and they're afraid to miss a minute of the action. So exciting for them...so exhausting for you!
Big hugs, and hang in there. I promise that it will get better. I can't say that my son magically turned into a great sleeper - even now, at age 7, he has trouble drifting off - but at least he is now old enough to stay in his bed and stay quiet even if he's awake.
Hoping for sweet dreams for all of you soon.