I don't know about poetry specifically, but what about ballads, which are really just popular poems set to music? There have been lots of studies published about ballads in the Tudor period and how the reflect concerns about events of the day. I feel sure there must be a few that contain references to people dying of "the sweat." If you have access to a larger university library (student or live near one), you might try checking their catalogue for ballad collections.
Have you looked at Berdan's, 'Early Tudor Poetry, 1485-1547'? It's available on Google books. Not sure if there is a great deal of information within this text but it might be starting point after you work through the ballads PhD Historian suggests.
Two authors you might want to check out: John Skelton, onetime tutor to Henry VIII, and court laureate. Right period, court association, but a narrow range of subject matter. Examples: "The Tunning of Eleanor Running", about a very unsanitory home brewing establishment, and "Why Come Ye Not to Court", a lengthy slam of Cardinal Wolsey. You could try reading or at least scanning all his writings in the hope of finding an allusion to epidemic disease.Thomas Nashe: Elizabethan, not Henrican, but has a few poems specifically on death by epidemic disease. The most famous goes by a variety of titles, including "A Litany in Time of Plague". "Brightness falls from the air, Queens' have died young and fair... The plague full swift goes by, I am sick, I must die, Lord have mercy on us." Published in a collection of prose and poetry called "Summer's Last Will and Testament", a title alluding to Henry VIII's (and Elizabeth's) fool by craft, Will Somers.
All comments are moderated so your replies may not show up immediately. Please be patient. Thanks!