The City of London is the name of an area of London, where the stock markets are etc. London is a city, but the actual "City of London" is an area between East and Central London. The City of London...
If you are walking then it is 0.7 miles, because you would go over Victoria Embankment. However, if you are travelling by car or taxi then it is 1.5 miles, for you must cross the Waterloo Bridge.
I think it used to be the city Jail. Now its a museum type thing, and you can go there and see the crown jewels on display etc....Provided I don't have my places muddled up.