The City is part of London under any normal person's interpretation of that word. Being as charitable as possible, it is true that the ceremonial county of Greater London does not contain the City. But the Greater London administrative area does.
It's also not "secret". Many people aren't aware of it, but that's through ignorance on their part rather than secrecy on the part of the City. The City plasters its logo all over the place: on all street signs in the City, on police uniforms, on statues, etc. Even the bollards in the City bear the colors of the City's flag. It's impossible not to notice that the City is different from the rest of London.
So that's two mistakes in the headline alone.
> The mayor of Greater London has nothing to do with the original City of London, which still has separate governing bodies and a mayor.
This is wrong. The Mayor of London has authority over the City of London, through the Greater London Authority (although some power is delegated to the Corporation, in the same way it is to borough councils).
There is indeed also a mayor of the City (confusingly called the Lord Mayor of London), but that mayorship is largely ceremonial in nature.
> Oddly enough, if the monarch wants to enter the City of London, she first must ask the Lord Mayor for permission.
This is just an urban myth, no such permission is required. In fact, the sovereign has precedence over the Lord Mayor (in the context of formal events and such like).