To be “flat broke” is to have absolutely no money. The word “flat” is added, perhaps, to emphasize that there is no bulge in any pants pocket to show even a single coin. “Flat broke” has been cited in print since at least 1840.
 
   
(Oxford English Dictionary)
broke, adj.
Freq. with qualifying word, as broke clean, broke dead, flat broke, stone-broke, stony broke.
1842 Spirit of Times 2 Apr. 58/1   Barrett, poor fellow, is dead broke.
1842 Spirit of Times 21 May 138/1   Every friend of Old Whitenose would have been flat broke!
1843 Spirit of Times 14 Jan. 544/3,  I was clean broke in less than four hours.
1846 Spirit of Times 25 Apr. 101/2,  I unfortunately am short of funds, flat broke, busted, collapsed.
 
23 September 1840, Ohio Statesman (OH), “Bank Swindlers in Mississippi,” pg. 4:
You must not presume from the above, that I am absolutely “flat broke,” and troubled with the blue devils.
 
18 November 1842, New Orleans (LA) Picayune, pg. 2, col. 5:
“I’m flat broke!” as the flatboat said when striking on a snag.
 
14 February 1844, Lorain Republican (Elyria, OH), “A Texian Hero,” pg. 1, col. 4:
About the time the Texas excitement ran so high in the United States, Jim Wills was in Pittsburgh, in that situation so common to play actors, viz: “flat broke.”
   
Google Books
The Mysteries and Miseries of New York:
A story of real life
By Ned Buntline
New York, NY: W.F. Burgess
1849
Pg. 36:
No; I am broke, flat broke! I haven’t a dollar left!”