Todd Park, the former chief technology officer of the United States, and Mikey Dickerson, who led a team of 60 engineers at Google and supervised the crew that fixed the Healthcare.gov website last year. Since that time, Park and Dickerson have been steadily recruiting an elite digital corps—a startup team, essentially, built mainly from the ranks of top private-sector companies—and embedding them within the U.S. government. Their purpose is to remake the digital systems by which government operates, to implement the kind of efficiency and agility and effectiveness that define Silicon Valley’s biggest successes, across everything from the IRS to...