I'm pretty sure that as a kid in the mid-1960s, going to school at Tower Hill School further along Chamber Street, that this building was a garage used by Post Office Telephones, the forerunners to today's BT.
For a number of years now, it has been a car wash.
Records in the London Metropolitan Archives relating to the Sun Insurance Office show that in 1807, number 14 Chamber Street was occupied by William Hegley who carried out the trade of Gun Maker. Research on Hegley indicates he was a manager at the gun-making business of Daniel Goff, who had been Master of the Gunmakers Company, London. It appears that when Goff died in 1840, Hegley took over the contract for the supply of guns to the East India Company.