Verbal irony is present in the setting of the story. The name card on the letterbox for the apartment that is the location for most of the action in the story reads "Mr. James Dillingham Young", a distinguished name once upon a time, but no longer. Now,
the letters of "Dillingham" looked blurred, as though they were thinking seriously of contracting to a modest and unassuming D.
As the story continues, it becomes apparent that the financial situation has gone beyond "thinking seriously" of moving to a less expensive status - Jim and Della are living on an extremely tight budget.
The comparisons of Jim's watch with the riches of King Solomon and of Della's hair with the beauty of the Queen of Sheba's possessions could be considered situational irony - the labeling of common items as being fabulously precious possession.