However, the Dassault Mirage III, the Lightning and the Starfighter weren't the only choices. Other choices included the Saunders-Roe SR.177, the SAAB J-35 Draken, the Convair F-102 Delta Dagger, the Convair F-106 Delta Dart, the Republic F-105 Thunderchief, the Vought F8U Crusader, the Grumman F11F-1F Tiger, and (according to some sources I've seen) the Sud Trident III.
Also, I'm not certain why the fact the Mirage wasn't in service matters that much. When the decision was announced in November of1958 the Lightning wasn't ready. The first Lightning wouldn't enter service until July of 1960, over 4 months after the superb Saab Draken entered service. Conversely, versions of the F-104 had been in squadron service service since 20 Feb 1958, the F-8U in April 1957 (or December 1956, depending on how you figure), and the F11F since late 1956 (or early 1957, again depending on how you figure).
I really doubt that the large, twin engined Lightning price equipped would've been $1.7m (in 1960 $USD) if the F-104G was $1.42. I think it's probably be 2x times that given the Lightning's complexity and Lockheed's comparative advantage in economies of scales.
Also, there was also US mutual aid/military assistance money, and other US government largesse. Plus training aid, plus who knows what else.
A really big reason for the F-104 being chosen was the technology transfer package Lockheed . This included having F-104 airframes and engines made in Germany by German companies.