"The Day Lady Died" is an elegy to Billie Holiday. An elegy is a poem of mourning and lament for someone who has died. Some of the most famous elegies in the English language are Thomas Gray's "Elegy Written in a Country Churchyard" and Walt Whitman's "When Lilacs Last in the Dooryard Bloom'd." O'Hara's elegy, though, departs radically from tradition, because the poem on the face of it doesn't seem to be about Holiday at all. It never mentions her by name, and she is described only briefly in the final lines.
The thing about writing a poem during your lunch hour is that you don't really have time to plan for some elaborate formal scheme. You write the words that come to you. That's what O'Hara has done here. Except for its division into twenty-nine lines and five stanzas, "The Day Lady Died" has none of the familiar markers to let us know we are reading poetry. No rhyme scheme or regular meter. There's an internal rhyme in line 3 between "1959" and "shoeshine," but it sounds unintentional. The poem has no punctuation and almost all of the lines are enjambed, meaning they are clauses that carry over to the next line.
You could think of the poem as consisting of three separate sentences with periods to divide them. The first stanza is a sentence, the second stanza is a sentence, and the last three stanzas form a long sentence. These are all run-on sentences that your English teacher would never let you get away with. They are cobbled together with the help of frequent uses of the word "and," making the speaker look busy, rushed, and possibly stressed-out. The most notable "literary" device is the capitalization of brand names. Check out "Symbols, Imagery, Wordplay" for more on that.