“The black gentleman” was a euphemism for the devil:
black gentleman n. Obsolete (with the) the devil.
Oxford English Dictionary.
The OED notes several similar euphemisms, including “the gentleman in black”, “the old gentleman”, and “the old gentleman in black”.
Austen also used the phrase in one of her letters to her sister:
What a contretemps! in the language of France. What an unluckiness! in that of Madame Duval.† The black gentleman has certainly employed one of his menial imps to bring about this complete, though trifling, mischief.
Jane Austen (8th February 1807). Letter to Cassandra Austen. In Edward Knatchbull-Hugessen, ed. (1884). Letters of Jane Austen, volume 1, p. 326. London: Richard Bentley.
† An allusion to the novel Evelina (1778) by Fanny Burney, in which the character Madame Duval laments, “But look here, now, here’s a cloak! Mon Dieu! why it looks like a dish-clout! Of all the unluckiness that ever I met, this is the worst!”
As for why the devil should be a byword for impatience, this is because patience is an important Christian virtue (for example, it is one of the seven capital virtues: chastity, temperance, charity, diligence, kindness, patience and humility), and so its opposite is associated with the devil.
And, beloved brethren, that the benefit of patience may still more shine forth, let us consider, on the contrary, what mischief impatience may cause. For as patience is the benefit of Christ, so, on the other hand, impatience is the mischief of the devil; and as one in whom Christ dwells and abides is found patient, so he appears always impatient whose mind the wickedness of the devil possesses. Briefly let us look at the very beginnings. The devil suffered with impatience that man was made in the image of God. Hence he was the first to perish and to ruin others.
Cyprian (3rd century). ‘On the Advantage of Patience’. Translated by Robert Ernest Wallace (1869). In Alexander Roberts & James Donaldson, eds. (1869). Ante-Nicene Christian Library: Translations of the Writings of the Fathers Down to A.D. 325, volume 13, p. 34. Edinburgh: T. and T. Clark.