Rodgers left his job at Scottish side, Celtic to replace Claude Puel who was sacked last Sunday after 16 months in charge.
He becomes the third manager to coach the pair of Ndidi and Iheanacho at the club after Craig Shakespeare and Puel and the fourth permanent manager in 23 months.
The former Liverpool and Swansea manager signed a contract until June 2022.
Former Arsenal, Liverpool, Manchester City, and Celtic defender Kolo Toure also joins Leicester as a first-team coach, having worked as a member of Rodgers’ backroom staff at Celtic.
Rodgers will be Leicester’s fourth permanent manager in 23 months after Puel, Craig Shakespeare and 2015-16 Premier League-winning manager Claudio Ranieri were all sacked.
The Foxes are 12th in the Premier League having lost five of their past six league games and host Brighton Tuesday (later tonight).
Saturday’s 4-1 defeat by Crystal Palace meant Leicester have lost four consecutive home Premier League games for the first time since January 2000.
They have conceded the first goal in 19 Premier League matches this season – more than any other side.