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:

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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!