I'm a fan of the "you don't have to go to sleep, but you need to stay in your room" approach. We bought lights that are easy to turn on and off so that each of our kids could turn the lights off on their own. DS (now 5) wasn't quite 2 when we started that. He used to look at books and would eventually fall asleep. DD (3 in Feb) tends to lie there staring off into space , and sometimes we'd find her asleep in the middle of her floor. When that would happen, we'd pick her up and put her in bed, but that hasn't happened in a while. I know there were times when she was out of bed, but she not only made it back into bed to go to sleep, but she tucked herself in as well.
What does your DD sleep in? DD hated her crib at a very early age (she only slept in it a handful of times...), so we switched to a floor bed. It made the whole "tucking herself in" possible, and it seemed to make things more comfortable for everyone, most importantly: her. She still cries every so often about an hour after she's been asleep. I go into her room, hug her and let her know that I'm there, and I ask her if I can stay for a few minutes. That calms her down quickly for some reason. If five minutes pass and she's still awake, I'll tell her that my few minutes are up, but I'll be back to check in on her. I do follow up on that, and sometimes she's still awake when I do, so she knows that I really do come back.
At bedtime, I concede on things that are inconsequential to me, for instance, for some reason it's a big deal for her if she can keep her door open. We can let that happen in our house (although I know not everyone's situation is the same as mine), so I let her feel that control over the bedtime routine by asking her if she'd like to keep her door open. Sometimes she'll ask before I do, and regardless, the answer is yes, though I am the one who determines how far open it can be. When we're done reading our bedtime books, I then ask her, "Do you want me to stay with you for one more minute or for two more minutes?" (The choices could easily be 5 or 10 minutes, but I became greedy with my me-time a while back.) She gets to feel control by getting to choose, and I'm only committed to one or two minutes. A few times she answered with "Two... HOURS!" (lol), but I responded with my usual, "you can choose one minute or two minutes. If you can't choose, I'll choose for you." Both DD and DS know how that usually goes. On the nights when she seems like she might be troubled with my impending departure, after she chooses the two minutes (she always chooses two minutes), I'll ask her if she'd like to hold hands too, which she does. One of the things I learned with all of this is that once the choices have been made, I need to do my part and hold her to those choices so she understands what her decisions mean. After two minutes, we're done. She's almost always awake when I leave her room and sees that when she chooses two minutes, she gets two minutes. If she's upset, I'll tell her I'll be back in 10 minutes to check on her, that she doesn't have to go to sleep, but she needs to stay in her room. We did this a lot at first.
One last thing... DS was an amazing napper. He could take two naps and still be in bed at his normal time and wake up at a good time the next morning. DD... oh no. No no no. We learned early on that powering through and skipping naps is the best course of action for her (and us). Yes, somedays she'll be cranky in the evening, but it's totally worth it compared to taking that nap and struggling at bedtime (which is between 7:30 and 8). That said, there have been times when she put herself down for naps (once I found her in my own bed tucked under the covers and everything), but usually when that happened, bedtime was surprisingly uneventful. I suppose she knew she needed the rest.
Best of luck to you with all this. Sorry for the long post. Every kid is different (even within the same family!), so I hope you find what works for you and your DD!