It landed the business after a pitch, which was called last month and involved a number of London and regional shops.
BJL will now develop a fresh positioning and relaunch strategy to reinvigorate the Silentnight brand.
An initial campaign is expected to launch next year, working across a number of media channels.
Adrian Fawcett, the chairman of Silentnight, said: "We loved the BJL team's way of thinking and strong campaignable thought. We look forward to working with this agency to see how its ideas come to life."
For the past few years, Silentnight has worked with creative agencies on a project basis only. Its last retained shop was Leith London, but that relationship ended around four years ago.
Carat Edinburgh, formerly Feather Brooksbank, handles Silentnight's media planning and buying account.
Silentnight's best-known advertising campaigns were created by TBWA in the 80s and 90s, featuring the brand's long-running hippo and duck characters.