"The woman sleeps on the bed."
Translation:Kvinden sover i sengen.
"På sengen" would be "on top of the bed", in this case. Like, "I left the suitcase on the bed" would be "Jeg efterlod kufferten på sengen". When you go to bed, you are "i sengen".
DaneTRs answer makes me wonder if there are regional differences to this. Are you Danish, DaneTR? :)
In any case, both "på" and "i" are accepted now, since they're both correct translations of the English sentence.