Unfortunately in this instance, that is where the rhyme will have to end, because he did die. It would have been nice to add the spider, bird, cat and all other animals to the rhyme, but he never got past the fly.
It is far more common for a Pope to die while in office than to resign, and when they do pass away it is usually due to either old age or illness. Occasionally though a Pope will meet their end in other circumstances. Pope Adrian IV, born Nicholas Breakspear, was just one of them. While enjoying a nice glass of wine one day in September, he swallowed, or nearly swallowed a fly and died.
Pope Adrian IV was the first, and only English born Pope. He became Pope on 4th December, 1154, following the death of Pope Anastasius. He reigned for 5 years before his untimely death on 1st September, 1159.
While sources from the time state that Pope Adrian IV death was due to a fly, some scholars believe he died from a complication of quinsy, which is a complication of tonsillitis. While this explanation is more reasonable and believable, it is still certainly possible that a fly was the catalyst to his death.