How do I make the solution in my own essay, which is based upon "A Modest Proposal," sound humorous or satirical?
This is a great assignment you are doing as it will help you to analyse the strategies that Swift employs in his own modest proposal that makes it so successful and powerful. The best place to start is to think about your own tone as the proposer of your solution. One of the ways that Swift makes his essay so powerful is that he actually assumes the voice of a economic planner who is very careful to back up his solution with a number of facts. He comes across as compassionate, caring and sensitive to the situation in Ireland. This of course makes his abominable proposal all the more shocking. Notice how he tries to present himself as a fair and thoughtful man before presenting his solution:
As to my part, having turned my thoughts, for many years, upon this important subject, and maturely weighed the several schemes of other projectors, I have always found them grossly mistaken in their computation.
Note how Swift only comes to his conclusions after "maturely weighing" the ideas of others. Phrases like this serve to boost his credibility before the actual unveiling of his not so modest proposal. Adopting a similar strategy by establishing a similar tone as a writer will help create the satire you are after. Good luck!