OED

The Online Etymology Dictionary has a citation a little earlier than the OED:

Phrase everything but (or and) the kitchen sink is 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," Jan. 24, 1944]

Antedatings

I found some earlier quotations, including the variant everything but the kitchen stove, that show the phrase predates World War II.

Everything but the kitchen stove can be found back to 1913 and appears more common in the 1910s than kitchen sink from 1914. In fact, I found just three sinks in Chronicling America compared to some 17 stoves.

Chronicling America only goes up to 1922. Trove and Papers Past show both variants were used in Australia and New Zealand before WWII, and in the Manchester Guardian in the 1930s.

Here's three of each.

Everything but the kitchen stove

The evening world., November 17, 1913, Final Edition, Image 3:

All of which is prefatory to saying that the Horse Show is just about the most distinguished assemblage of well-dressed people that these United States can produce. Of course some of the women have on everything but the kitchen stove (this is being written by a man) and some of the men bat less than .003 when It comes to a coat or a waistcoat but, as a gathering of thousands, the Horse Show sessions afford one an opportunity to see not only the last cry but the very best taste in personal adornment, masculine and feminine.

The Washington times., June 19, 1914, HOME EDITION, Image 3:

He completely lost his head in the sixteenth century, and wore everything but the kitchen stove, and today--yes, long suffering women, rejoice--today he promises to array himself once more in those colors which stamp him unmistakably and irrevocably as the VAIN MALE.

The sun, October 18, 1916, Page 13, Image 13, boxing headline:

DILLON UNABLE TO DROP TIM O'NEILL
Hits Him With Everything Except Kitchen Stove, but Celt Lasts Limit.

Everything but the kitchen sink

The Washington Times (Washington D.C.), February 20, 1914:

Having "blown in" his savings on a complete new set of scenery, Jerry was logged out this day like a circus horse. He had on everything but the kitchen sink and the door mat.

A advert for Macy's in the New-York Tribune, February 19, 1919:

Pots and Pans!

To say nothing of rolling pins, clothes baskets, wash boilers, percolators, casseroles, I towel bars, cloth ventilators, china, cut glass, earthenware -- well, in fact everything but the kitchen sink is included in this

New-York tribune., August 27, 1922, Page 2, Image 62, "Better Late Than Never" by W.E. Hill:

"Well, peoples, the first hundred years is the hardest, they say! We should worry." Howell is the original comic boy. Just LISTEN to him! Pretty soon he will be saying that Mrs. So-and-So had "everything on but the kitchen sink." Maybe you've guessed it by this time. Yes, Howell is one of those unfortunate jokers who never quite catch up with the current gibe and jest. Two years from now Howell will probably be talking on "Mad Money" and "Finale Hopper."

This suggests the phrase was already considered old hat by then, and it was at least 10 years old and had even been used in a Macy's ad.

Edit

Here's some even earlier stoves (and a range), from 1906 to 1914. All the early stoves (up to 1911) comment on a woman's attire. As far as I can tell all authors and publishers are American.

"The Other Doors" by Mary Heaton Vorse published in Scribner's Magazine - Vol. XXXIX, May 1906, No. 5 - Page 603:

But Felicia merely remarked, "She had on everything but the kitchen stove, and yet she looked well dressed!"

"In the Ballingers' Box" by Harold Susman published in The Smart Set - Vol. XXX - No. 4 - April, 1910 - Page 66:

ALGIE She has on everything but the kitchen stove. And now that I look closer, I see that she has that on, too!

(Another possible 1910 or 1911 but GB is snippet only and I can't find this volume at Hathitrust or Internet Archive:

the lady "had on everything but the kitchen stove" in anticipation"

)

Prince Or Chauffeur?: A Story of Newport by Lawrence Perry - A.C. McClurg & Company, 1911 - Newport (R.I.) - Page 335:

He was vaguely amused at the remark of a woman beyond the first bloom of youth, who, turning to her companion and nodding toward a socially famous young matron, who preceded them down the stairs fairly jingling with jewelry, remarked:

"I say, Jerry, Mrs. Billy has put on everything but the kitchen stove."

It confirmed in Jack's mind an impression which had begun to form, that the smart set, so-called, is not altogether lacking in, well,—smartness.

This is not a comment on a women being dressed up to the nines, but rather taking almost everything on a journey. Also "kitchen range" rather than "stove".

"An Adventure in Contentment" by George Palmer Putnam - Outdoor World & Recreation - Vol. XLIX - September, 1913 - No. 3

Such was the invariable wail at a long portage, inevitable the world over, for trim down equipment as heroically as you will, and yet it seems as if you had about everything but the kitchen range, and the grand piano when it comes to back- packing. We had little enough; no tent, a sleeping bag each, and the proven fundamentals of the culinary department, plus the satisfactory luxury of camera equipment.

This is also not to do with a woman's clothes, but another "almost everything".

A chapter letter for "Louisiana Alpha-Newcomb College" by Mildred Post in The Arrow of Pi Beta Phi - Volume 30 - March 1914 - Issue 3 - Page 468:

It was the last school day before Christmas, and we had the usual Christmas tree with a present for the room from each member, besides a present for each girl from one other girl. We actually could not say that the presents were "everything but the kitchen stove," for even the "kitchen stove," at least a very diminutive one, was included for one girl. But the "eats" were wonderful and we had just the very best time we could have had.