"military engine for hurling stones," mid-13c., from Old French mangonel "catapult, war engine for throwing stones, etc." (Modern French mangonneau), diminutive of Medieval Latin mangonum, from Vulgar Latin *manganum "machine," from Greek manganon "any means of tricking or bewitching," said to be from a PIE *mang- "to embellish, dress, trim" (source also of Old Prussian manga "whore," Middle Irish meng "craft, deception"), but Beekes thinks it might be Pre-Greek. Attested from c. 1200 in Anglo-Latin.
machine for smoothing and pressing linen and cotton clothes after washing, 1774, from Dutch mangel (18c.), apparently short for mangelstok, from stem of mangelen to mangle, from Middle Dutch mange, which probably is somehow from to Vulgar Latin *manganum "machine" (see mangonel), "but its history has not been precisely traced" [OED].
The possession of a mangle, for the use of which a small sum was charged, is, among the poorer classes of English cottagers, a common means of earning money. The question 'Has your mother sold her mangle?' (quot. 1836-7) was at one time the commonest piece of 'chaff' used by London street-boys. [OED]