Complete Stylistic Analysis

The story under analysis is written by Edgar Allan Poe, a famous American author, poet, editor, and literary critic. The name Poe brings to mind images of murderers and madmen, premature burials, and mysterious women who return from dead. His works have been in print since 1827 and include such literary classics as “The Tell-Tale Heart”, “The Raven”, “The Fall of the House of Usher”. This versatile writer’s oeuvre includes short stories, poetry, a novel, a textbook, a book of scientific theory, and hundreds of essays and book reviews. He is widely acknowledged as the inventor of the modern detective story and an innovator in the science fiction genre, but he made his living as America’s first great literary critic and theoretician. Poe’s reputation today rests primarily on his tales of terror as well as on his haunting lyric poetry. 

This story may not take place in a morgue or a creepy house, but it wouldn't be Edgar Allan Poe without a touch of the Gothic. Here, it's the simple, smoky darkness of the "little back library, or book-closet, au troisième, No. 33 Rue Donot, Faubourg St. Germain".
Au troisième means that their apartment is on the third floor, which is probably actually the fourth floor, because in France the first floor is considered the ground floor and not counted. 

While the dark, smoky room creates a Gothic mood, it also seems definitely anti-Gothic. People talking, thinking, and smoking pipes in a library seems stable, calm, and not very scary or creepy. There's even something comforting about it. It's a setting perfect for the coldly logical explanation of how Dupin solved the mystery.

Dupin and the narrator have already been sitting in the dark, thinking, for an hour before  Monsieur G— arrives. When he shows up, Dupin decides not to turn on the lights because G— needs him to think about something: "If it is any point requiring reflection […] we shall examine it to better purpose in the dark".
This is a little bit weird, because it's a common thing to associate knowledge with light - "Knowledge is light and ignorance is darkness". But according to Dupin, all of life's truths are in plain sight. To see them we sometimes need to obscure everything except what we are examining.
When Dupin goes to D—'s house to look for the letter, it's actually daytime. Still, he closes his eyes, figuratively, to everything but the letter, even using green glasses to darken his vision (like sunglasses). In contrast, G— visits D—'s apartment only at night. The darkness is supposed to provide cover for him to look for the letter.
So: Dupin uses darkness in light; G— uses light in darkness.
Ultimately, the story suggests that seeing clearly or not seeing clearly have little to do with darkness or light, and lots to do with individual vision, and individual ability to process what is seen. So, the setting of "The Purloined Letter" plays with ideas of light and darkness.
The plot has an intrigue, what is quite okay for the detective stories. 

Two men sit in the dark, silently smoking. C. Auguste Dupin, private detective, and his roommate, the unnamed narrator, are puffing away in a smoky reverie when G—, the head of the Paris police, enters the scene.
G— spends a lot of time rather cryptically explaining the mystery. Basically, D— has purloined a secret and scandalous letter, which belonged to an unnamed royal lady. He's now blackmailing her to get what he wants politically, and she's asked G— to purloin it back.

 Every night for the past three months, G— has been searching D—'s hotel room for the letter. Thoroughly. How thoroughly? Try this:

We examined the rungs of every chair in the hotel, and, indeed, the jointings of every description of furniture, by the aid of a most powerful microscope. Had there been any traces of recent disturbance we should not have failed to detect it instantly. A single grain of gimlet-dust, for example, would have been as obvious as an apple. Any disorder in the glueing —any unusual gaping in the joints —would have sufficed to insure detection.
All this, and still no luck. Meanwhile, the political situation isn't improving. So G— showed up at Dupin's door, asking for help.

A month later, G— show up again still without a letter. Surprise! Dupin has it now, and sells it to G— for fifty thousand francs. G— gets what wants and Dupin shows off his massive brain and brilliant intellect. At the very end, Dupin reveals that he purloined the letter because he digs the royal lady on a political and possibly personal level, and wants to get revenge on D—. 

There are  4 main characters in the short story: C. Auguste Dupin,

Unnamed narrator, Monsieur G——, Minister D——.

C. Auguste Dupin -  A savvy and learned Parisian who helps the city’s police to solve crimes. Dupin uses psychology to ruin the plans of a thief and find a stolen letter that the police of Paris could not find by conventional investigations. Dupin solves the theft of the letter by putting himself at risk politically. Whereas the Paris police tread lightly around the actions of Minister D——, an important government official, but Dupin ignores politics. In this story, Dupin reveals his capacity for revenge. When the Minister insulted him in Vienna years before the present crime, Dupin promised to repay the slight. This story demonstrates that Dupin’s brilliance is not always dispassionately mathematical. He cunningly analyzes the external facts of the crime, but he is also motivated by his hunger for revenge.

Unnamed narrator -  A friend of Dupin. In awe of Dupin’s brilliance, the narrator faithfully recounts Dupin’s explanations without doubting or challenging him.

Monsieur G——  -  The Prefect of the Paris police. Limited by his conventional police rules, Monsieur G—— depends on Dupin’s assistance in peculiarly difficult crimes, and his own general competence highlights Dupin’s superior abilities.

Minister D——  -  A government official and the thief of the letter. Minister D—— ‘s ability to outwit the police in his crime proves, that he is a worthy adversary for Dupin.

The story under analysis is rather big, that is why E. A. Poe used a lot of different expressive means and stylistic devices. 
To describe the setting, the main characters and generally to describe the development of the events in the story, the writer provides epithets: "For one hour at least we had maintained a profound silence.", "We gave him a hearty welcome; for there was nearly half as much of the entertaining as of the contemptible about the man, and we had not seen him for several years.","and this fact gives the holder of the document an ascendancy over the illustrious personage whose honor and peace are so jeopardized.", "Than whom," said Dupin, amid a perfect whirlwind of smoke, "no more sagacious agent could, I suppose, be desired, or even imagined.", " Do you not see he has taken it for granted that all men proceed to conceal a letter, --not exactly in a gimlet-hole bored in a chair-leg --but, at least, in some out-of-the-way hole or corner suggested by the same tenor of thought which would urge a man to secrete a letter in a gimlet-hole bored in a chair-leg?"

The story is rich in similes. They contribute to the description of the characters actions or states: "We gave him a hearty welcome; for there was nearly half as much of the entertaining as of the contemptible about the man, and we had not seen him for several years. ", "The thief," said G., is the Minister D--, who dares all things, those unbecoming as well as those becoming a man.", "Entirely," said the Prefect. "He has been twice waylaid, as if by footpads, and his person rigorously searched under my own inspection."

To stress and to draw the readers attention to something important the writer uses inversion: "Yes," replied the Prefect; "and the power thus attained has, for some months past, been wielded, for political purposes, to a very dangerous extent.", "Had the letter been deposited within the range of their search, these fellows would, beyond a question, have found it.", "Do you not see he has taken it for granted that all men proceed to conceal a letter, --not exactly in a gimlet-hole bored in a chair-leg --but, at least, in some hole or corner suggested by the same tenor of thought which would urge a man to secrete a letter in a gimlet-hole bored in a chair-leg?"

Polysyndeton makes the text more rhythmical and contributes to the vivid description of the setting: "For some minutes he remained speechless and motionless, less, looking incredulously at my friend with open mouth, and eyes that seemed starting from their sockets; then, apparently in some measure, he seized a pen, and after several pauses and vacant stares, finally filled up and signed a check for fifty thousand francs, and handed it across the table to Dupin.", "When I wish to find out how wise, or how stupid, or how good, or how wicked is any one, or what are his thoughts at the moment, I fashion the expression of my face, as accurately as possible, in accordance with the expression of his, and then wait to see what thoughts or sentiments arise in my mind or heart, as if to match or correspond with the expression."

Asyndeton in it's turn makes the utterances more dynamic: "This functionary grasped it in a perfect agony of joy, opened it with a trembling hand, cast a rapid glance at its contents, and then, scrambling and struggling to the door, rushed at length unceremoniously from the room and from the house, without having uttered a syllable since Dupin had requested him to fill up the check. " 

Let's not forget about the allusion, which is not as obvious for us, as it was for Edgar A. Poe : "A certain set of highly ingenious resources are, with the Prefect, a sort of Procrustean bed, to which he forcibly adapts his designs."