Women were not constables at the turn of the century. They formed their own police service in 1914. The 1st female officer wasn't sworn in until 1919...but still couldn't carry handcuffs or arrest people until 1923.
At Houdini's party, an actor playing the King of England is seen. In 1900, when the episode is set, Queen Victoria was still reigning. King Edward VII didn't become king until January 1901.
When Doyle is reading the ledger, two dates are visible: 19 October 1900 next to Lucy's, and 8 December 1900 next to Daisy's name, possibly noting their time of death. the episode takes place six months after Lucy's death, so April 1901 the earliest.
When Holmes and Doyle are discussing their theories of the case, Houdini dismisses Doyle's statement with "Garbage in, garbage out." That phrase didn't come into use until the advent of the computer age, more than five decades after the time frame of this program. It's highly unlikely that Houdini would have invented that phrase, given the absence of evidence that anyone picked it up from such a celebrity and used it in the intervening years.
The Metropolitan Police didn't get their first woman police constable until 1919. They didn't get their first woman detective constable until 1973. Adelaide would never have been employed by the Met in 1901.