One example of irony in this story is the revelation at the end of the story that Mrs. Purdy knew that the woman her lodger was looking for died in the same room that he had been occupying. This is an example of dramatic irony, whereby the reader knows something that one of the characters does not, or, in this case (given that the lodger is now dead), did not. This is also a rather tragic irony, as the lodger had, just previous to this moment in the story, been desperately ransacking and destroying his room in an effort to find even some small trace of the woman.
There is also a tragic kind of irony in the similarities between the deaths of the lodger and the woman he was searching for. The lodger had been looking for this woman for five months. These months he described as "Five months of ceaseless interrogation and the inevitable negative." The lodger then came agonizingly close to finding the woman, only to miss her by one week. At the end of the story, the lodger, desperate and hopeless, committed suicide in the room by turning the gas "full on" and laying "gratefully upon the bed." We then find out that the woman he had been searching for also committed suicide in the same room, one week prior to the day, also by turning on the gas. This indeed is a rather macabre, grisly, and tragic irony.