To me, the major irony in this story takes place at the end of the story. It has to do with the way that the various people react to what they have just done.
What is ironic, to me, is that they are pleased that the hanging went well. It would have been bad, they say, if the man had been dangling, not dead, and have to have his legs pulled to help kill him. But they do not talk about this as good for the sake of the dead man -- they are saying it is good because it makes it easier for them, the executioners.
To me, this is the irony -- they are not really concerned with the person they have just killed. Instead, they are concerned with themselves and how the execution affects them.