I live in Ealing next to Wembley

and I am not English so a bit ignorant on those subjects
Sylvain, please read the following post for the reason why Middlesex is still in use now
I have to say I was surprised yesterday when I read the comments stating Wembley was not located within the London area, when I new for 100 percent that it was located within one of the London Boroughs.
For your information, the county of Middlesex was disestablished as far back as 1965, with the legislation coming into effect nearly 45 years ago, as the articles I linked to explained in great detail.
I assume what caused the confusion for you was the fact that the Postal County for areas in Greater London was not altered following the 1965 reforms.
Basically the changes to Middlesex came about following a Royal Commission on Local Government in Greater London in the early 1960s. The Parliamentary Act, The London Government Act 1963, recognised officially the conurbation known as Greater London and created a new local government structure for the capital.
The Act came into effect on 1 April 1965, but despite the disappearance of the county, Middlesex has routinely been used for postal addressing due to the postal county having been retained because Royal Mail were unable to follow all the changes to county boundaries, which was reported as being due to it being deemed too expensive at the time.
I live in one of the London Boroughs to have previously been located in the Middlesex country boundary and we still to this day write Middlesex for postal address purposes, even though Royal Mail officially ceased using counties in the direction of mail from 1996.
Treble99 is correct in saying the legacy of Middlesex exists to this day with a number of organisations based in the old area of Middlesex using Middlesex within its name such as Middlesex County Cricket Club and Middlesex University.
As I said previously, and will repeat again, Wembley is in north-west London, and is part of the London Borough of Brent. Brent is one of the 32 London Boroughs.
Middlesex exists today only as a postal county. As I explained in the above post, Royal Mail did not implement the changes in the 1960s due to the huge cost involved. You live in the
London Borough of Ealing, not Middlesex County Council!!!