"Cry-it-out" and "sleep training" are concepts based in parents' desperation and exhaustion. CIO can work well and quickly for a few littles, but having experienced/watched outcomes in three generations now, it seems to me to this approach fails more often than not. Many babies and toddlers are just not good sleepers. It's how they are put together.
Tired children would sleep if they could. But sleep can be tricky, especially in the face of big changes in the child's life, or anything that causes pain. Leaving a child crying in the night is not "sleep training" – it seems to me it only teaches a child to get tough and keep her suffering to herself. Helping set up the conditions where sleep is more easily accessible is far better, if slower.
I clearly remember lying awake for hours at night feeling cold or thirsty or scared or dealing with some discomfort. Calling for Mommy brought silence, harsh words or even a whack on the butt. I don't think this servedme or my sisters well, and having gone through those difficult baby-sleep months with my daughter and my grandson, I do have deep empathy.
There are dozens of influences that can make it harder for kids to sleep. Becoming over-tired or over-stimulated, or not getting enough physical exercise, make falling and staying asleep elusive. Growth spurts can by highly stimulating for a week or three at a time. Too much evening exposure to light, especially in the blue/green part of the spectrum interferes with the brain's ability to manufacture melatonin, our natural sleep hormone. Too little bright daylight or exercise in earlier hours can be a huge problem for some children.
A once-unrecognized problem is becoming more visible lately: exposure to the many odorous or scented chemicals in household cleaners, air fresheners and laundry products can stimulate the nervous system and push relaxation and sleep out of reach (and cause many other problems).
Some peds recommend giving children additional melatonin, but it's a powerful hormone, so please do ask your doc before dosing. Meanwhile, try to boost daytime physical play, especially outside; limit screen time two hours before bed, lower lights and noise levels in the evenings.
My grandson was a terrible sleeper until 2.5, and then he magically transformed into a sleeping champ. It was a tough time for the adults in his life, but we were pretty certain he couldn't help it. His parents tried to let him cry by himself a few times, but it didn't "take" for any of them. They all found far more comfort in comforting him when he needed it.
As he got older, they set up a reward system. If he was calling them, say, 3xnight, they put three pennies in a small bowl by his bed, and removed one each time they went to him. If he still had any pennies left in the morning, he would get a favorite treat with his breakfast; extra jelly on his toast or a slice of bacon. He caught on quickly and after a few nights began choosing to deal with his wakefulness himself. He started playing or looking at books until he fell asleep again. I don't think he was able to do this before the age of 2, though.
Hang on, M., once this stage passes, you'll probably be surprised at how fast your daughter grew up. Here's a really informative website on kids' sleep. You may find information here that will help you http://www.med.umich.edu/yourchild/topics/sleep.htm