St Martin, Ludgate, is an Anglican church on Ludgate Hill in the ward of Farringdon, in the City of London. St Martin, Ludgate, also called St Martin within Ludgate, was rebuilt in 1677–84 by Sir Christopher Wren.Wikipedia
I have always found this church open during the week. On most mondays between 1pm and 2pm there are music recitals.
Friends’ Watchers present from 11am to 3pm on Fridays.
Nearest stations: City Thameslink, St Pauls and Blackfriars.