Translation:The mother has been putting her daughter to bed early.
"Put to sleep" is a common euphemism for "euthanize". I have always put my children and grandchildren "to bed", but not always early. I agree with Lingodingle that children are like body parts and, for that matter, bedrooms. The definite article stands in for a possessive adjective appropriate to the subject of the sentence.
I sent a report about this about a week ago. The Oxford Dictionary states pretty clearly that "put to sleep" can only mean "euthanize" or "anesthetize" http://www.oxforddictionaries.com/definition/english/sleep?q=put+to+sleep#sleep__27 Glad I'm not the only one who finds this annoying!
From New England. I believe that we use "put to bed" much more frequently than "put to sleep" in the context of "getting the kids into bed". I associate "to put to sleep" with (1) euthananizing a pet or (2) in the expression: That book/lecture is so boring that it will put you to sleep.
In UK English, put to sleep is what you do to very old and sick dogs and horses. That is you euthanise them.