David Miller didn't always make razors... in 1829-33 he was an ironmonger and table knife maker with premises in Orchard Street; then he moved to Fargate and operated a warehouse. In 1852 he set up as D. Miller & Son (Joseph) at the Eldon Works in Eldon Street.
The makers mark was a broom, but this was acquired by T. H. Blake after D. Miller & Son ceased operating, then by Castle & Turton.
David died in 1870 and Joseph in 1880. The firm seems to have continued - in 1883 it was situated at the Premier Works in Wentworth Street. Although it is said to have finished operations in 1893 another company of the same name is recorded as being in operation in Carver Street from 1911 until sometime in the 1920's.
Regards,
Neil