John Edwards ended 2004 with a problem: how to keep alive his public profile without the benefit of a presidential campaign that could finance his travels and pay for his political staff. Mr. Edwards, who reported this year that he had assets of nearly $30 million, came up with a novel solution, creating a nonprofit organization with the stated mission of fighting poverty. The organization, the Center for Promise and Opportunity, raised $1.3 million in 2005, and — unlike a sister charity he created to raise scholarship money for poor students — the main beneficiary of the center’s fund-raising was...