Mother has been sick and hasn't gotten any sleep for the past three days.
Literal
Mother [topic-は] these three-days sick [because-で] is-not-sleeping.
Two tricky pieces. First, 病気で is the て-form of 病気だ used to link clauses with a loose 'being sick (and/so)' sense — not the instrumental で. Second, 寝ていません is the negative of ~ている: 'has not been sleeping,' a perfect-continuous negative that says the action has failed to occur across the stated span. この三日間 = 'these past three days' with 間 marking the duration.