Well, you have clearly identified the situational irony that occurs in this excellent short story, however, you are also right in suspecting that other forms of irony exist in this story. Remember, that irony can be detected whenever there is a gap between appearance and reality, and therefore, the ending of the story, when the narrator intervenes to give us his opinion of these two "foolish children". Note what he says:
The Mgie, as you know, were wise men - wonderfully wise men - who brought gifts to the Babe in the manger. They invented the art of giving Christmas presents. Being wise, their gifts were no doubt wise ones, possible bearing the privilege of exchange in case of duplication. And here I have lamely related to you the uneventful chronicle of two foolish children in a flat who most unwisely sacrificed for each other the greatest treasures of their house. But in a last word to the wise of these days, let it be said that of all who give gifts, these two were the wisest. Of all who give and receive gifts, such as they are wisest. Everywhere they are wisest. They are the Magi.
There exists irony then in the fact that in the eyes of the world, what Della and Jim did was foolish, and yet the narrator tells us that they are "the Magi" - their spirit of sacrificial giving out of love makes them closes to the original Magi who brought gifts to the Jesus-child at Christmas all those years ago. Although the world would scorn them for what they have done, they show they know more about giving than anyone else.