"Hell's Kitchen" is no borrowed name like "Chinatown" (from San Francisco) or "Soho" (from London). According to Irving Lewis Allen, City in Slang (1993), pg. 233: "Hell's Kitchen, at least as a name pertaining to a neighborhood, is possibly indigenous to New York."
15 August 1881, New York Times, pg. 5:
The tramps on the West Side congregate in two dilapidated buildings knownas "Hell's Kitchen" and the "House of Blazes," in West Thirty-ninth-street, near Tenth-avenue.
2 March 1882, Ohio Democrat (New Philadelphia, OH), pg. 4:
Popular Names in New York.
The city has a variety of localities whose names sound strange to unfamiliar ears, and hence are among the peculiarities of the metropolis. There is "Mackerelville," and also "Cowboy," and many other spots noted in the police reports. Among the more notorious is "Hell's Kitchen," which was brought before the public by the Rooney murder. "Hell's Kitchen" is a series of rookeries inhabited by a class deeply sunk in misery, but notwithstanding poverty they always have money for whisky. Rooney only beat his wife to death -- which is so common a thing here that it hardly calls for more than passing notice. "Hell's Kitchen" is no worse than "Murderer's Row" or "Devil's Eelpot," each of which has its record. Then, too, look at the fanciful names enjoyed by an important element in our population. There are the "Short Boys," the "Dead Rabbits," the "Man Eaters," and others, who not only vote, but often control elections, and also do their share to fill the penitentiary. "Hell's Kitchen" is a richly remunerative investment to the landlord, for the poor always pay highest rent, and the tenements of this city are immensely profitable. This class of property holders claim respectability, and may live in elegant style, while their wealth is wrung from the most miserable of mankind. -- New York Letter.