The verbal irony in this passage turns on the narrator's misinterpretation of what "persons of a desponding Spirit" mean when they express "concern" about the large numbers of older, diseased and disabled poor people dying in Ireland. What his friends mean, of course, is that they are morally concerned and distressed over the situation and would like the narrator to come up with a plan to end the suffering of the poor by making their lives better. The narrator, however, interprets his friends' "great concern" as wanting to get rid of the poor because they are an economic drain on the system. He interprets their "despond" as the result of the fact that the poor are not dying fast enough.
To the narrator's mind, the people who approach him with this question are unnaturally depressed--ie "desponding Spirits"--because actually there is no problem! As the narrator points out, there's no need for a scheme to rid the country the burden of the poor because they are dying off so fast anyway. The narrator, who defines the "matter" wholly in economic terms, says "I am not in the least pain upon that matter... [for] they are every Day dying, and rotting, by cold, and famine, and filth, and vermin, as fast as can be reasonably expected." The irony, of course, is that the fact that old, sick and disabled people are dying and rotting in cold, hunger and filth is exactly what should cause pain in the narrator's mind. Likewise, the narrator finds the younger workers in a similarly "hopeful condition:" so starved that if they do happen to get work, they can't do it and die anyway, thus "happily" solving the problem of how to get rid of them.
The verbal irony lies narrator's inability to see the poor in anything but economic terms, so that he misinterprets people's "concern" for them not as compassion for their plight, but as an economic desire for the poor to die faster. Thus, the narrator turns morality itself on its head, describing as "hopeful" and "happy" a morally disgraceful situation in which the poor are dying in cruel, horrible and preventable ways. In doing this, Swift hopes to shock people into treating the poor as human beings deserving of empathy and compassion.