It first appeared in print in 1948 in a dictionary of military slang. "They chucked everything they'd got at us but - or sometimes 'including' - the kitchen sink" was a typical way for a soldier to describe a heavy enemy bombardment.
I would guess that it came from the days before fitted kitchens, when most houses had Belfast sinks and due to their weight, that when you moved house you took with you 'everything but the kitchen sink'.