Dick, an experienced counter-terrorism officer, was the first woman to lead London’s 193-year-old police force, also known by the name of its Scotland Yard headquarters.
Interior Minister Priti Patel had extended her contract in September.
Dick said Everard’s murder and “many other awful cases recently” had damaged confidence, but the force had turned its full attention to rebuilding trust.
Khan singled out Dick’s reaction to the report on the behaviour of 14 serving police officers at Charing Cross Police for making her position untenable.
“The response from the Commissioner wasn’t up to the scale of the change required in the Met Police Service,” he told broadcasters.
Prime Minister Boris Johnson said on Twitter that Dick had “served her country with great dedication and distinction over many decades”.
Patel, who will be responsible for choosing Dick’s replacement, said earlier this month there were problems with the culture of the force.
On Thursday, however, she said Dick had undertaken her duties “with a steadfast dedication to protecting our capital city and its people – including during the unprecedented period of the pandemic”.
The choice of her successor has political ramifications because the Met is investigating parties at the centre of Johnson’s government during the pandemic for breaches of lockdown laws.
Khan, who is in the opposition Labour Party, said he could not comment on the parties’ investigation because it was an operational matter.
But he said he recognised the huge public interest in allegations because Britain policed by consent.
“Intrinsic to policing by consent is the public having confidence in our police officers,” he said.
Hours before Dick had called into a BBC radio show to vow to saying she had “absolutely no intention” of standing down.