Perhaps you have to that question all figured out, but I like to revisit, see what authors who worry this genre every day have to say about it. Death after all is a natural event; it comes to us all. As the doctor in Unnatural Death remarks, specific time of death is unpredictable--no one can really know the day, the hour, the minute death will take hold. (It's rather like predicting the exact moment of a natural birth.)
Sayers poses this problem as follows: The "victim" is old and terminally ill with cancer but in good spirits despite her pain and gradually weakening powers. She is surrounded by capable and beloved caregivers, and preparing to "let go." If this person dies a bit unexpectedly, should one suspect murder? Well, maybe--poor old auntie did argue with her solicitor towards the end and was a bit paranoid that her hired nurse was trying to poison her. So perform a postmortem. But what if this autopsy turns up nothing amiss? No sign of poison. No marks of murder. Lots of old people get a bit paranoid towards their ends. And more importantly, what does it matter one way or another if death is hurried by a couple of months? Has a real crime been committed? And if so, how? By whom? Not surprisingly, Lord Peter's imagination is captured by this "pretty problem" and we, the readers, are off . . .
Lord Peter's gut, of course, tells him a murder has been committed. His friend Inspector Parker's experience however says not. We are treated to their musings on how easy it is to murder someone; how frequently murderers are not only not caught, but also no even suspected. I googled that, and found--lo!--only about half the murders in the United States even today are solved. H'm. That's not good. And that's only for those deaths officially recognized to be murders. For we have no Lord Peter on the scene, with his good humored doggedness following out his instincts. Who knows how many "natural" deaths are in fact "unnatural"?
I could go on. Suffice it to say, this one is fun. The plot is nicely complicated; more suspicious deaths occur. Possible perpetrators appear and disappear. But the method remains unresolved until the end, when a tricky little event in Parliament (!?) provides the key. Sayers hits her stride I think, as authors do, around book three or four. Lord Peter has matured. He's still very attractive and very witty, but not quite so silly as in the first two novels, and if you found the upperclass lisp a bit annoying, it moderates herein. Looking forward greatly to The Unpleasantness at the Bellona Club, which comes next.