When you're trying to describe the style of a poem, imagine how you would describe it if it were a person. The style of Jane Austen's poem "I've a Pain in my Head" is very similar to her prose style: satiric and sly.
It is a narrative poem, which means it tells a story. Actually, the story could be considered a joke as well. A "suffering" lady named Beckford asks her doctor, the "dread" Newnham, what to take for a headache. His reply? "Ah, what can you do Ma'am?" Not much of a doctor, right? Beckford comes up with her own solution, and the doctor is in such agreement, he decides he needs the same medicine. This is funny in two ways. First, it satirizes the medical profession, which must have been very unhelpful in Austen's time. Second, it makes fun of hypochondriacs, who probably do give their doctors headaches.