I do think that learning independent sleep is important and most definitely a factor in why you're having so many sleep issues. I also think that overtiredness might be playing a role. However, I know tons of kids whose parents used an attachement parenting approach to sleep (basically what you have done, by first nursing, then rocking, then lying with her to get to sleep), and I have not heard of one who sleeps as poorly as your DD does now. I'd honestly say I think it's more than just a lack of sleep training, because thousands upon thousands of kids are never sleep trained and still sleep much, much better than this. I would definitely push for a sleep study myself. It's true that some kids learn to sleep well later than others, but almost all are sleeping through the night or close to it (maybe one night waking) at age 2.5 years.
First prong, I think you should start by teaching independent sleep. I think the gradual withdrawal approach where you sit next to her while she falls asleep and then move the chair in increments until you're out of the room is the nest way to go. When she wakes at night, go right back to the chair until she falls asleep. Every three nights, move the chair further away until you're out the door. If you don't think it will mean nap doesn't happen at all, then try it for nap at the same time, but if it's going to be a battle, just deal with the bedtime and nightwakings until you're ready to tackle naps.
I also would move bedtime earlier, to 7 pm, and start nap no later than noon, possibly as early as 11:30, given her early waking. Once she's sleeping better, the early waking might take care of itself. It might not. I have to say I'm no expert in this, because I've so far been blessed on that score, so check out the early waking support thread on this board for help with that.
{{{HUGS}}}. You must be utterly exhausted.
