The 27-year-old was a key part of the Foxes side that won the Premier League in 2015-16, but asked to leave the club amid interest from the Blues.
Chelsea have also signed defender Davide Zappacosta from Torino on a four-year deal for an undisclosed fee.
However, striker Fernando Llorente opted for Tottenham instead of a move to Stamford Bridge.
And Everton midfielder Ross Barkley changed his mind about joining Antonio Conte's side.
Leicester expect to confirm the signing of Adrien Silva, 28, from Sporting Lisbon on Friday for £22m as a midfield replacement.
Drinkwater played in 35 of Leicester's 38 league games in their title-winning season and formed an impressive midfield partnership with N'Golo Kante, who joined Chelsea last July.
He has yet to play this season because of a thigh injury.
The playmaker, who joined Leicester in 2012 from Manchester United, has won three England caps since making his international debut in March 2016.
"I'm delighted to be a Chelsea player and can't wait to get started," said Drinkwater. "It has been a long journey to get here but I am very happy and am looking forward to helping the club win more trophies."
Zappacosta has been capped four times by Italy, with Blues boss Conte handing him his first call-up to the national squad in May 2016.
The 25-year-old full-back joined Torino from Atalanta in 2015 and made 29 Serie A appearances last season.
"I hope I can show my quality to the coach, my team-mates and the supporters," said Zappacosta.
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Chelsea have signed midfielder Danny Drinkwater from Leicester City for £35m.