Several good points made by people on both sides of the argument about 'LL & 'KK.
However, the FACT remains that the customer demand & strategic solution, is for ONE very much larger Hub & an uninterrupted amount & frequency of flights from it.
IF you build a 2nd. Runway at 'KK that's fine for 'KK. But it only gives you another airport the same size as the present 'LL. It does NOT provide what is needed ie the very much larger ONE main international gateway.To do that would require 2 or 3 extra runways at 'KK & a very much larger terminal.
Without the Thames Estuary development, the expansion should , logically, take place at Heathrow.
Two similarly sized airports (at 'LL & 'KK) are not what will meet the requirement. Both will be too small compared to what is needed.