How far behind the USSR was the USA after Sputnik?

The two Sputnik satellites of 1957 were themselves of little military significance, and the test missile that launched them was too primitive for military deployment, but Khrushchev claimed that long-range missiles were rolling off the assembly line "like sausages," a bluff that allowed President Eisenhower's opponents--and nervous Europeans--to perceive a "missile gap." Khrushchev in turn tried to capitalize on the apparent gap in a series of crises, but his adventurous policy only provoked perverse reactions in China, the United States, and Europe that undermined his own political support at home.

Eisenhower was apprised in advance of Soviet missile progress thanks in part to overflights of the U-2 spy plane. By the time of Sputnik the Pentagon already had several parallel programs for ballistic missiles of various types, including the advanced, solid-fueled Polaris and Minuteman. The great fleet of B-47 and B-52 intercontinental bombers already deployed also assured continued American strategic superiority through the early 1960s.

4 Months, until we matched them..

"In response to the surprise launch of Sputnik I on October 4, 1957, the U.S. restarted the Explorer program, which had been proposed earlier by the Army Ballistic Missile Agency (ABMA). Together with the Jet Propulsion Laboratory (JPL), ABMA built Explorer I in 84 days and launched it on January 31, 1958. Before work was completed, however, the Soviet Union launched a second satellite, Sputnik 2, on November 3, 1957."