Correct me if I'm out of date but doesn't London, indeed the City of London, already have a dedicated GA and Business airport in Biggin Hill which is significantly better placed that a few other (supposed) London airports that I could mention.
Despite all the mocking of Manston's poor location Manston has/had significantly better road and public transport facilities than Lydd has, and may ever have.
Even if KLM were doing OK with yield on their Manston operations I'd suggest that Lydd management have long since approached KLM regarding Lydd operations, alas Lydd was specifically built to serve the cross channel market pre ferry and chunnel services, it's location is specifically to serve France pretty much making it useless to serve anywhere other than south of itself.
As for runway extension etc. ... What for? ... Manston tried and failed with a based low cost carrier operation, not even the save Manston group are suggesting that a re-opened Manston would have a viable future with passenger operations, a KLM service alone wouldn't finance the costs of a new runway and terminal at Lydd and any IT charter operations in/out of Manston clearly didn't last if at all.