Which poetic device is used in the sentence (there was an ickyness, a sinking, a sickening of the heart.)

The answer is: alliteration.

Alliteration is a literary device which consists of the repetition of similar consonant sounds in stressed syllables in nearby words.

In the sentence from "The Fall of the House of Usher," alliteration can be found in the following words: iciness, sinking and sickening. In that respect, the recurrence of sounds intensifies the tone of the passage, in which the narrator provides a description of his friend's ruined house.