“Fudge Factory” is a nickname of the U.S. Department of State. The term was coined by Washington (DC) Post syndicated columnist Joseph Kraft (1924-1986) in 1965-1966. “Fudge Factory” was probably influenced by the term “fudge factor” (“fudging"something into a desired result).
 
The nickname “Fudge Factory” has also infrequently been used to describe the Pentagon.  The Central Intelligence Agency has been called the “Pickle Factory” since at least 1965.
   
 
Wikipedia: United States Department of State
The United States Department of State (often referred to as the State Department or DoS), is the United States federal executive department responsible for international relations of the United States, equivalent to the foreign ministries of other countries. The Department was created in 1789 and was the first executive department established.

The Department is headquartered in the Harry S. Truman Building located at 2201 C Street, NW, a few blocks from the White House in the Foggy Bottom neighborhood of Washington, D.C. The Department operates the diplomatic missions of the United States abroad and is responsible for implementing the foreign policy of the United States and U.S. diplomacy efforts.
 
The Department is led by the Secretary of State, who is nominated by the President and confirmed by the Senate and is a member of the Cabinet. The current Secretary of State is Hillary Clinton. The Secretary of State is the first Cabinet official in the order of precedence and in the presidential line of succession.
   
(Oxford English Dictionary)
fudge, int. and n.
b. An act of fudging (in sense Additions d); an unsatisfactory or makeshift solution, esp. one reached for the sake of expediency. Also, sophistry, prevaricatory or imprecise language or reasoning. Chiefly in political contexts.
1980 Observer 22 June 5/1   The right wing of the party believes that, in concocting his fudge, Mr Callaghan has sold the pass.
1981 Guardian 10 Sept. 12/1   Given the choice between fudge and mudge and a firm statement of its views on incomes policy the TUC can usually be counted upon to come out clearly for ambiguity.
1990 Economist 22 Sept. 72/2   Unwillingness to put up with political fudges.
 
fudge factor n. a factor speculatively included in a hypothesis or calculation, esp. to account for some unquantified but significant phenomenon or to ensure a desired result.
1977 Aviation Week 24 Jan. 62/3   Meeting criteria for the FAA, jet transport A/C do not land on runways based on actual rollout distance. Thrust reversing is not included and many fudge factors are added.
1983 Nature 21 July 210/1   Bohr’s boldness in formulating what would now be called fudge factors was vindicated: by formalizing the unexplained, Bohr set up the challenge that was eventually met when the entirely new physics of quantum and wave mechanics was expounded.
1989 Financial Times 22 Feb. 22/7   The market soon recognised the fudge factor—half a point tacked on for the drought effect—and settled back into more familiar expectations on growth.
 
(Historical Dictionary of American Slang)
fudge factor n. an allowance for imprecise calculation; margin for error; (also, joc.) a mathematical factor required to convert an incorrect answer to a problem into a correct answer.
1962 (cited in Webster’s 10th)

Fudge Factory n. Pol. the U.S. Department of State. Joc.
1983 R. Thomas Missionary 270: I’m retiring from the fudge factory in exactly two months and nine days.
1986 Former U.S. Asst.Secy. of State B. Kalb (public lecture, Knoxville, Tenn.)(Nov.18): The building that is known everywhere affectionately as the Fudge Factory. 
 
Wikipedia: Joseph Kraft
Joseph Kraft (4 September 1924—10 January 1986) was an American journalist.
 
After working at the Washington Post and the New York Times in the 1950s, he became a speechwriter for 1960 Presidential candidate John F. Kennedy. His work landed him on the master list of Nixon political opponents. He served as one of three panelists for the third and final debate, held at Phi Beta Kappa Memorial Hall at the College of William & Mary, in the 1976 Presidential Election.
 
Kraft was a graduate of Columbia University. His syndicated column ran in over 200 papers.
 
28 November 1965, Salt Lake Tribune (Salt Lake City, UT), “Bundy’s Future Beclouds Role of His Staff System” by Joseph Kraft, pg. 32A, col. 6:
He wanted to consider himself sharp alternatives before they had passed through the fudge factory of inter-agency compromise.
(...)
The Bundy staff, in these circumstances, was an absolute necessity. It reached down and brought alternatives to the President’s attention before they had been erased in the fudge factory.
 
21 May 1966, Arizona Republic (Phoenix, AZ), “Dean Rusk Runs A Fudge Factory” by Joseph Kraft, pg. 7, col. 7:
The fact is that the department has not been run primarily as a decision-making instrument. It has been run as a fudge factory. The aim has been to make everybody happy, to conciliate interests, to avoid giving offense and rocking the boat.
 
Google News Archive
1 October 1966, Toledo (OH) Blade, “Ball Leaves State Dept. Post; Only Welles Topped Tenure,” pg. 2, col. 1:
Mr. Ball (George W. Ball—ed.), 56, still seems to exude the same boundless energy he has displayed since his entry in 1961 into what he jokingly terms “the fudge factory” where foreign policy is made.
   
28 October 1966, Trenton (NJ) Evening Times, “Job A Tafy Pull, Says Katzenbach,” pg. 8, col. 8:
WASHINGTON (UPI)—Some have called the State Department “a fudge factory.” But after 24 days as the department’s No. 2 man, Nicholas deB. Katzenbach feels it’s more like a taffy pull.
   
13 January 1967, Chicago (IL) Tribune, “The Fudge Factory,” pg. 16:
The state department, dubbed by John F. Kennedy a “bowl of jelly” and by former Undersecretary George W. Ball a “fudge factory,” is now engaging the agonized attention of Nicholas de B. Katzenbach, Ball’s successor.
 
Google Books
The State Department;
11 articles
By David K Willis
Boston, MA: Christian Science Pub. Society
1968
Pg. 36:
The three men, all halfway up the promotion ladder at Class 4, work in what George Ball once referred to as “the fudge factory” and Arthur Schlesinger Jr. calls a “bowl of jelly.”
   
Google Books
The Center;
People and power in political Washington
By Stewart Alsop
New York, NY: Popular Library
1968
Pp, 119-120:
It seems far more likely that, like his predecessors, Katzenbach will sooner or later abandon the Sisyphean task of trying to defudge the fudge factory, and turn to the greener pastures of policy-making.

Google Books
The Atlantic
Volume 222
1969
Pg. 8:
When George Ball resigned from the State Department in 1966, he suggested that he might write a book called something like “My Years in the Fudge Factory.”
 
Google Books
The Foreign Affairs Fudge Factory
By John Franklin Campbell
New York, NY:  Basic Books
1971
Pg. VII:
I know of no other book that describes the way the foreign-policy bureaucracies of Washington—the “fudge factory,” if you will—really functions.
 
19 September 1971, Washington (DC) Post, pp. D1-D2:
Fantasies from the Fudge Factory
By William L. Givens
 
Google Books
Community Consultation
By Patrick O’Neill and Edison J. Trickett
San Francisco, CA: Jossey-Bass
1982
Pg. 44:
The State Department was so ponderous and bureaucratic that it was referred to as the “fudge factory” by its detractors.

Google Books
Bright Star
By Harold Coyle
New York, NY: Pocket Books
1991, ©1990
Pg. 11:
As he wandered along the seventeen and a half miles of corridors of the Pentagon, affectionately known as the Fudge Factory, Major Scott Dixon couldn’t make sense of the excitement generated by the report of a training accident.

Google Books
The Silent Service:
Los Angeles Class
By H. Jay Riker
New York, NY: Avon Books
2001
Pg. 34:
Most simply called it (The Pentagon—ed.) “the Building,” but those in a more critical mood referred to it as “the Fudge Factory,” “Fort Fumble,” “the Squirrel Cage,” or — Gordon’s favorite by far — “the Five-Sided Wailing Wall.”
 
Google Books
Safire’s Political Dictionary
By William Safire
New York, NY: Oxford English Dictionary
2008
Pg. 784:
In Washington, the State Department is called “the fudge factory.”