] I didn't realise it had been totally swallowed up by London.
] It is a borough or something now, rather than a County?

I am not aware of it having any legal existance, although it is true that some people in West London still use the name on addresses etc. Why they should is a bit of a mystery as the Middlesex filter beds face the Essex filter beds, over the river Lea at Hackney and no one in East London claims to be living in Middlesex! I guess it is just that the West London part was the last to go.

Of course there are the sporting clubs, most famously Middlesex County Cricket Club, and I did my jury service in Middlesex County Court which is in Parliament Square. There is even a campaign to re-instate the county!