Now it is 10 pm and you put your daughter on the bed and tuck her in and you want her to fall asleep quickly.
Now your daughter sit up and talk and play with toys on the bed.
I see this in Oxford dictionary
[intransitive] get to do something to reach the point at which you feel, know, are, etc. something
After a time you get to realize that these things don't matter.
You'll like her once you get to know her.
Can I say to my daughter "quickly get to sleep" or "quickly fall asleep" or "go to sleep" in this situation?
I don 't think "go to bed" is suitable because she is already in bed.