The Angel, Vine Street - unfortunately this pub seems to be closed at the weekend. It is possible that it may open in the Summer when the tourists arrive in their droves. The closest other pub that I found was the Three Tuns on Jewry Street, but this too was closed. If both these are closed the only alternative is to try the next pub you find on the way to Liverpool St Station.
A second alternative is the other Vine street in Central london, Bond street area, but once again we found no pub so just filled in with a spare along the way.
Note: Look who's examing the paperwork?
If you're feeling lazy you can take the tube from Aldgate to Liverpool St, one stop. Otherwise, walk through subway and head up Houndsditch.