L. - your little Brooklin needs a bedtime routine that is all her own... one that makes her room and her bed a special place. Here is how to make this happen:
Pick out CHAPTER books that are calm but interesting: Charlotte's Web, Winnie the Pooh, Francis, Little Bear. Tuck her in and sit by her bed, or IN it with her, and read aloud, slowly and softly.
Show her the pictures if you want to but don't let her touch the book - this is story time and she will create the pictures in her head.
She will fall asleep before you are done reading for the night, of course. So pay attention to her while you read so that you remember where you left off. Since she will start out in her bed, over time she will just remain there through the night.
Then, make a trip every week to the library to seek out new and better bedtime stories, giving you something to do for fun when she is awake and giving her MORE attachment to her room and her bed, since these bedtime stories are just for her.
If she doesn't fall asleep before the story is over (or after one chapter), you can keep reading or sing her a sweet bedtime song. Mom's bedtime songs are the best! If you feel you can't sing, bring a tape/cd player into the room and put on quiet word-free music. She'll love falling asleep with the music and the images of her new book friends in her head.
Also, get her into the habit of choosing one of her favorite stuffed animals or dolls to listen to the story with her in her bed so that she has a story buddy. She may change animals/dolls with every story, or may choose the same one for an eternity. This will be the last thing she will do before climbing into bed... knowing that her friend is missing the story, and possibly sleeping alone on the cold, hard, living room floor will give her a feeling of importance and reverence.
Now you have put HER in charge of someone else who NEEDS her in order to have their own good bedtime experience.
Just to be clear, in the end, you may have to stay with her until she is fully asleep, regardless of how long it takes, to ensure that she feels safe and stays in her bed.
LOVE, not punishment, works with two year olds. Reading and singing a child to sleep is the loving way to ensure that she feels safe.