Johnson introduced the London rental standard in 2012 as part of a plan to accredit good landlords and letting agents, with the scheme eventually coming into force in 2014.
However, since then, the scheme has accredited less than 2,000 new landlords, despite setting a target of 100,000 by 2016.
Papers released by Khan show that Johnson ignored repeated internal advice on the scheme, including a May 2012 briefing that it would take more than 50 years to reach the 100,000 target.
Khan said: “We know most landlords do a good and decent job – but we’ve got to stop those rogue landlords who are exploiting renters. I’ve got plans to take them on, but it’s very disappointing that little or no progress was made under the previous Mayor.
“It seems that Boris Johnson was asleep at the wheel and ignored repeated warnings from officials that his plans were not working.”
The mayor's comments came as part of a visit to a Hackney support service run by homelessness charity Shelter.
Shelter chief executive Campbell Robb said: “Millions of Londoners are stuck in a 'rent trap' - paying tens of thousands of pounds for a home they can never call their own. At the sharp end there are thousands of homeless families in London right now – over 2000 in Hackney alone – trapped in temporary accommodation because there’s nowhere they can afford to live.
“It doesn’t have to be like this. The new Mayor of London has the power to turn our housing crisis around, by building homes that ordinary Londoners can actually afford to rent or buy. We welcome the Mayor’s commitment to prioritise building genuinely affordable homes for Londoners, and look forward to working with him to deliver on this pledge.”