From Heathrow Airport:

Underground direct to Earls Court Station.

From Gatwick Airport:

Take the Gatwick Express to Victoria station, form there take the underground (District line, Westbound) to Earls Court station.

From Stansted Airport:

Take the Stansted Express to Liverpool St Station, from there take the underground (Circle line, Eastbound) to Gloucester Road Station. Once outside station take a left and walk down, turn left on to Cromwell Road and walk down, cross over to the otherside of the road, once you past Sainsbury supermarket take the first right and you are on Lexham Gardens. 

From Earls Court Underground:

Take Earls Court Road Exit, cross the road at the Traffic lights and walk down Hogarth Road, turn left at the end of the road you will now be on knaresbrough road, at the end of the road you will see a set of traffic light cross over, Lexham Gardens is between Quality crown hotel and majestic Hotel