"Schlaft schon" is proper German but sounds a little odd, too. We would rather say: "Schlaft endlich" (sleep finally). My grandfather from Brno who grew up with Brno-German as his mother tongue would say "Schlaft schon" to us as children. So the czech somehow influenced the German.
It seems that the adverb 'already' has mutated into a completely new type of word that instructs time travellers to go back in time and do something in the past. Languages change but I do not think that I will be using this.
Yes, I am an old fart that should 'get over it already!'.
As a matter of interest the latest edition of Fowler's modern english usage 'already' used in this way means yet, still or even now but its use is not recommended. I remember from my time in south africa jewish people added 'already' to sentences as a meaningless addition. Despite the kind reply by karl I still do not understand what meaning this czech sentence is trying to convey.
My 3rd comment on this controversial topic! I feel a translation into British English should be allowed, but we Brits should accept the American as correct too. It isn't always the British grammar that came first anyway, some old English words and grammar have survived longer in America than in Britain.