I also feel your pain! My son, now 2 1/2, started waking around 5:30 just after his first birthday. He was in a crib in our room and no amount of back rubbing, singing, rocking, holding, anything, would get him back to sleep.
The first thing we had to overcome was the teething. Every single one of his teeth caused extreme discomfort and a bad cold. It was such a pain, but after all those big teeth finally came in we moved him into his sister's room (only a 2 bedroom house) where he slept slightly better but would still wake most mornings at an unreasonable hour.
We tried sleep training a few times (didn't work), earlier to bed, shorter nap, nothing made a difference. In the end, we would bring him into our room when he woke so he wouldn't wake his sister, and I would nurse him and on the lucky mornings when he fell back asleep I'd carefully slide down into bed with him and he'd sleep on my chest.
I never thought I'd be so accommodating to a young child for the long term! But, damn, waking up at 5:30 sucks. My husband and I still trade sleeping in a little every other morning and while it hasn't been great for our marriage, it gets us through and it's getting better. You have to come up with some compromise that works for your family. I keep telling myself he won't be doing this when he's 4 or 5.
Nowadays, he feels his nap is "optional" and so we have a different set of problems. He wakes at 5:30 a lot less, usually it's more like 6:20, but we have major nap struggles during the day.
Good luck. I wish I had a magic pill or spell I could send you. I'll think of you next time I'm up at 5:30.