Democrats were also experiencing a prolonged stretch of success in Congress, controlling the House for the vast majority of this stretch and the Senate only slightly less frequently- what were the Republicans major policy goals in this time? Has any other American party survived such a fallow stretch?
To understand Eisenhower's election, it's important to understand the twenty-or-so-years prior.
The Great Depression changed the political landscape of America. There has long been a debate about whether Franklin D. Roosevelt's New Deal brought the United States out of the Depression, but World War II definitely did, and the success and hope created made lifelong Democrats out of many voters. The programs implemented by FDR and the New Deal Congress were so popular that it became difficult for Republicans to argue against them; in addition, the laissez-faire administration of Herbert Hoover stained the Republican Party for years. It was very much the "wave the bloody shirt" method of the mid-20th century.
It is clear there was a division in the Republican leadership. The face of the party, often the nominees for president, were East Coast moderates. Men like Thomas Dewey, who was the party's nominee in 1944 and 1948, and Wendell Willkie, the nominee in 1940 (a former Democrat, in fact). These men were often internationalists and embraced the New Deal programs, promising to make them more efficient. On the opposite side of the spectrum were men like Robert Taft, Senator from Ohio, who led the conservative wing. Taft was a powerful figure in the Senate and opposed many New Deal programs, was the co-author of the Taft-Hartley Act, and was an isolationist. The true power of the party was in the moderate wing, however; Taft failed in his bid to become the party's nominee in 1940, 1944, 1948, and 1952.
1952 was seen as an opportunity for Republicans. President Truman was highly unpopular due to the Korean War and several corruption charges against his administration. Truman would have been allowed to run for another term because he was excluded from the 22nd Amendment's term limits thanks to a grandfather clause, but his age, desire to be home, and motivation from his wife Bess convinced him to stand down; he would subsequently lose the New Hampshire primary to Estes Kefauver, but he put up no meaningful opposition.
Eisenhower can be seen as a political general; a person doesn't ascend to the rank of Supreme Allied Commander without having political instincts. But following the end of World War II, he had kept himself out of the fray and did not share his political views. Eisenhower had to be sold on the idea of running for president, and once he was, he declared himself publicly as a Republican, though as it would become clear, he really fit more in the mold of Thomas Dewey than Robert Taft. This is a primary reason he was paired with Richard Nixon as his running mate, to appease the conservative wing with a hard-nosed, tough anti-communist with experience in government, as well as someone who was younger.
Republicans were "in the wilderness" so to speak from the White House, but their efforts to control one or both houses of Congress were neutralized, for the most part, because FDR and the spirit of the New Deal turned many people into Democrats. It wouldn't be until the turmoil of Lyndon Johnson and Jimmy Carter's respective presidencies that things would really shift in favor of the Republicans on local levels, which is how they would regain control of Congress.
Edit: Information on this topic was primarily taken from David M. Oshinsky's essay "Harry S. Truman" and David L. Stebenne's "Dwight D. Eisenhower" from the collection "The American Presidency" edited by Alan Brinkley and Davis Dyer. Additional info can be found in "Presidential Leadership" edited by James Taranto and Leonard Leo. Additionally, two documentaries I would recommend from PBS's American Experience series are "Truman" and "Eisenhower."
Has any other American party survived such a fallow stretch?
Ironically, the Democrats. Republicans held the Presidency for 59 the 72 years in between the elections of Lincoln and FDR (and one of the years for Democrats was Andrew Johnson switching parties in the last year of the term he inherited from Lincoln). In that same time, Republicans had a majority in the senate in 29 of of the 36 congresses (and a tie in one other). And had the House in 23 of those congresses.