I'm afraid you are showing how little you know about the governance of London.
Firstly, yes the London Fire and Emergency Planning Authority, which governs the fire service, is body within the Greater London Authority of which Sadiq Khan has been mayor for a year. I have no doubt that any inquiry into the fire will examine whether certificates, warnings, and advice from the fire service to the block management were correct, and whether there was anything about the service's response to the fire that could have been inadequate. If there were problems, then I'm sure the mayor will address them. Whether he would therefore be personally culpable for those mistakes is a very different matter. It would be unreasonable to suppose that if there are systemic failures on the LFEPA he would be likely to have identified them and put them right in the space of 12 months. Boris Johnson and Ken Livingstone both had 8 years to do so, yet Grenfell House still happened.
Second point,...is just wrong. The GLA is not responsible for council estates, the council is. In this case the Borough of Kensington and Chelsea, which owns the block and the TMO (Tenant Management Organisation) that was created to manage the block. The GLA plays no part in ownership, management or policy setting for the block. If you are wanting to find local government responsibility for this incident you need to look a Kensington and Chelsea and the leader of that council, Nick Paget-Brown.