sleep doula
n. A caregiver who teaches new parents how to get their child to sleep longer at night.
Examples
2007
Tracey Ruiz, 36, is a Toronto sleep doula, which means she does house calls and teaches parents how to get baby to go to sleep.

Parents call her "because they are exhausted," says Ruiz.
—Trish Crawford, “Doing it their way,” The Toronto Star, January 20, 2007
2007
Ruiz got into the sleep business while working as a postpartum doula. A doula is trained to offer women information and emotional and physical support during and after pregnancy and birth.

She formed Sleep Doula three years ago after hundreds of new parents complained to her about sleep deprivation.

Her technique relies on old-fashioned comforting techniques combined with some time allowing the baby to cry it out, which teaches the baby to self soothe. She also develops a feeding and sleeping program for mom.
—Denise Davy, “Sleepless in Mommyland,” The Hamilton Spectator, January 05, 2007
2006 (earliest)
Baby-induced sleep loss, observes Alija Fischell, a psychiatrist at Women's College Hospital, is clinically proven to push even the most even-keeled mother off her rocker, so why not introduce some kind of preventive measure to keep families thriving and together?

"I think the concept of a sleep doula is wonderful, and for some women it is absolutely necessary in reducing postpartum depression, psychosis or anxiety disorders. But for any woman, whether at risk or not, sleep protection is important, because what happens? Irritability. Anger. Outbursts. And who takes it? The family."
—Deirdre Kelly, “Mother's newest little helper: the sleep doula,” The Globe and Mail, November 04, 2006
Notes
A doula is a caregiver who provides prenatal, childbirth, and postpartum support. Doula is a Greek word that refers to a slave or, more accurately in this context, a woman who serves another woman.