The irony is the story is that Bertha has all along believed she has a perfect, blissful life, only to find she has been living in a fool's paradise. This occurs when she realizes her husband, Harry, is having an affair with another woman, Miss Fulton:
she saw . . . Harry with Miss Fulton's coat in his arms and Miss Fulton with her back turned to him and her head bent. He tossed the coat away, put his hands on her shoulders and turned her violently to him. His lips said: "I adore you," and Miss Fulton laid her moonbeam fingers on his cheeks and smiled her sleepy smile.
Ironically, Harry does not find his bliss in Bertha but in another woman.
As the above passage indicates, Mansfield uses the literary device of descriptive imagery to show rather than tell the reader what is happening. However, the key literary element that Mansfield uses throughout to bring about her irony is point-of-view. We see everything that happens through Bertha's eyes—it is her perspective alone that tells the story. No omniscient narrator ever interprets for us that in the scene above, for example, Harry is having an affair: it is up to us to figure that out from the description of what Bertha sees.
Keeping the point-of-view entirely in Bertha's hands allows Mansfield's irony to have a greater impact. Because Bertha does not question the bliss of her life, we do not either, for the simple reason that we have nobody to tell us that Bertha's perspective is skewed. This makes the impact when Bertha recognizes what is going on all the more forceful to us.