The new Metropolitan Police commissioner must tackle police culture and conduct, Priti Patel has said.
Writing in the Evening Standard, the home secretary said “strong and decisive new leadership will be required to restore public confidence”.
Dame Cressida said she had been left with “no choice” but to resign, after London’s mayor made it clear to her he had no confidence in her leadership.
It follows cases of sexism and misogyny among some Met officers.
Last week, the police watchdog found “disgraceful” examples of bullying and sexual harassment at Charing Cross police station in central London.
Dame Cressida, the first woman to lead the UK’s biggest police force, also faced criticism over the murder of Sarah Everard by a serving Met officer last year, and a series of other scandals.
Beyond London, the Met is also responsible for national counter-terrorism policing, and Ms Patel said the Met commissioner was “a national leader, with a critical national role”.
Dame Cressida is thought to have offered her resignation after declining to meet Mr Khan to discuss her plans for reforming the Met.
She said she had “agreed to stay for a short period to ensure the stability of the Met”.
Her successor will be appointed by the home secretary, in consultation with the mayor of London.
Last Friday, the Met submitted a plan to Mr Khan on reforming the force.
But Mr Khan said he was “not satisfied” with Dame Cressida’s response to the scale of change required to “root out” racism, sexism, homophobia, bullying and misogyny in the Met.