"My little sister did all of the exercises and went to sleep."
Translation:Моя младшая сестра сделала все упражнения и легла спать.
To go to sleep = to fall asleep They both mean уснуть. If you want to say легла спать then you should say "went to bed" (not "went to sleep").
For example, your child can be already lying in bed, but talking and not settling, so you can say the him or her "Hush baby, go to sleep".
уснула is the correct translation of "went to sleep".