"life at ease," by 1902 (as Reilly), popularized in U.S. during World War I; it seems to have been military slang initially, sometimes said to trace to various songs but none of that title has been found.
sink to wash food, dishes, etc., 1824. Phrase everything but (or and) the kitchen sink is attested from 1944, from World War II armed forces slang, in reference to intense bombardment.
Out for blood, our Navy throws everything but the kitchen sink at Jap vessels, warships and transports alike. [Shell fuel advertisement, Life magazine, Jan. 24, 1944]
Earlier was everything but the kitchen stove (1919).