NFL.com's Ian Rapoport helped fill in the details.
A team source told Rapoport that the Eagles learned at 10 a.m. ET Wednesday that they for sure had Kelly. The source said Eagles general manager Howie Roseman was the biggest factor in landing Kelly, playing the last 10 days "incredibly. ... No one was more prepared and aggressively pursued." Roseman's approach was similar to what a college coach would use to recruit a player: push, push, push, know when to pull away and give the person time to think.
The Eagles continued to sell Kelly on the people in their organization, even after he turned them down. They told Kelly he'd remain their top priority even as they interviewed other candidates. They previously sold to Kelly that they had two quarterbacks (Michael Vick and Nick Foles) who could run his system.
While it's an open question whether Vick or Foles fit what Kelly does, the Eagles' continued pursuit of Kelly clearly paid off.
Les Bowen of the Philadelphia Daily News wrote that Roseman first heard Kelly could be persuaded on Jan. 8, when Roseman was in Nashville, Tenn., at a coaching convention to meet with Notre Dame coach Brian Kelly. After Brian Kelly said no, the Eagles again touched base with Chip Kelly, and things heated up in the last 24 to 36 hours.