In 2004, the Republican master strategist Karl Rove led weekly sessions at his Washington residence where he planned the re-election of President George W. Bush. In April, Mr. Rove summoned several of the important players behind Mr. Bush’s ascendance to his home once again, this time to draw up plans to push a Republican resurgence. The landscape has changed, with Mr. Rove at times clashing with potent new Tea Party-style activists, some of whom view him as a face of the old party establishment they want to upend. Already a prominent presence as an analyst on Fox News Channel and...