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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Ben, from Sheffield, was 21 months old when he disappeared on the Greek island of Kos on 24 July 1991.
A team of 19 South Yorkshire Police officers, forensic specialists and an archaeologist are excavating farmland near to where he was last seen playing.
Members of the Hellenic Rescue Team and Red Cross are assisting.
Ben's family believe he was abducted, but police are now investigating whether he was accidentally run over and killed by a bulldozer.
As the search team began a seventh day of digging, Det Insp Jon Cousins read a statement from Ben's family praising the volunteers.
"We are so incredibly thankful for the help and support of the volunteer search teams working with officers," they said.
"To know that people are giving up their own time and are as desperate as we are to find the answers about what happened to Ben is something we will be eternally grateful for.
"We've been told that volunteers are coming to the site on their days off or straight from work, and we honestly can't thank them enough for that and for their dedication."

The 41-year-old has recorded 696 home runs, putting him fourth on the Major League Baseball all-time list.
He was given a 162-game ban for doping in January 2014, meaning he missed the entire 2014 season.
Rodriguez's final game will be at Yankee Stadium against the Tampa Bay Rays and he will then take on an advisor role at the club.
He helped the Yankees win the World Series in 2009 and was voted the American League's Most Valuable Player in 2003, 2005 and 2007.
"This is a tough day. I love this game and I love this team and today I am saying goodbye to both," Rodriguez said.
"This is also a proud day. I was 18 when I broke into the big leagues and I never thought I would play for 22 years.
"No player ends their career the way they want to, we all want to keep playing forever but it doesn't work that way. Accepting the end gracefully is part of being a professional athlete.
"I want to thank the fans for letting me play the game I love."
Chelsea have signed midfielder Danny Drinkwater from Leicester City for £35m.
[NEXT_CONCEPT]
The family of missing toddler Ben Needham say they are "eternally grateful" for the volunteers who have been helping with the police search.
[NEXT_CONCEPT]
New York Yankees slugger Alex Rodriguez has announced he will retire from the sport on Friday.