"a fight for it with the dusty miller"
by cm
The Sandman
Public DomainThe Sandman - Credit: Vilhelm Pedersen

Presumably "dusty miller" is intended to be a term for sleep, although we can find no other such usage.  A number of plants and pubs are named "Dusty Miller", and in the twentieth century the term was used as slang for a coal miner.

Curiously, Dusty was a nickname for men named Miller in the late nineteenth century, and "dustman" was a term for sleep (e.g. to have met with the dustman meant to feel sleepy), akin to the sandman.  It looks like Mina, or possibly Stoker, is getting a little confused.