What is the type of irony in "The Furnished Room"?

Much of the irony in "The Furnished Room" is verbal irony. It is to be seen in the contrast between the repellent appearance of the room and the whole building, on the one hand, and the humorous way the narrator describes the place, on the other. For example:

Upon the gay-papered wall were those pictures that pursue the homeless one from house to house—The Huguenot Lovers, The First Quarrel, The Wedding Breakfast, Psyche at the Fountain. The mantel's chastely severe outline was ingloriously veiled behind some pert drapery drawn rakishly askew like the sashes of the Amazonian ballet. 

The verbal irony is also glaring in the housekeeper's description of the furnished room and its actual condition.

“This is the room,” said the housekeeper, from her furry throat. “It's a nice room. It ain't often vacant. I had some most elegant people in it last summer—no trouble at all, and paid in advance to the minute."

We can imagine what sort of "elegant people" would stay in such a room and in such a building. Evidently the housekeeper, whose name is Mrs. Purdy, has lived in this dingy, decaying building for so long that it looks relatively attractive to her.
The ending of "The Furnished Room" contains a different sort of irony. Once we learn that the girl the young lover has been searching for had committed suicide in the same room he has just rented, we see the dramatic irony of the entire story. The girl gave up trying to survive as an entertainer in the cold, cruel city of New York, and her lover gave up searching for her in the setting where her life had ended only a week before. Both committed suicide with the same gas jet in the same furnished room, a room which seems to have swallowed both of them up and forgotten them.