The Lodger by Marie Adelaide Belloc Lowndes

A serial killer known as "The Avenger" is on the loose in London, murdering blonde women. A mysterious man arrives at the house of Mr. and Mrs. Bunting looking for a room to rent. The Bunting's daughter is a blonde model and is seeing one of the detectives assigned to the case. The detective becomes jealous of the lodger and begins to suspect he may be the avenger