Old English slæpan "to be or fall asleep; be dormant or inactive" (class VII strong verb; past tense slep, past participle slæpen), from Proto-Germanic *slēpanan (source also of Old Saxon slapan, Old Frisian slepa, Middle Dutch slapen, Dutch slapen, Old High German slafen, German schlafen, Gothic slepan "to sleep"), from PIE *sleb- "to be weak, sleep," which perhaps is connected to PIE root *sleg- "be slack, be languid," the source of slack (adj.). Sleep with "do the sex act with" is in Old English:
Gif hwa fæmnan beswice unbeweddode, and hire mid slæpe ... [Laws of King Alfred, c.900]
Related: Slept; sleeping. Sleep around first attested 1928.