The British phrase to talk the hind leg off a donkey means to talk with unflagging and wearying persistence or to have the power to persuade another by eloquent or charming speech (cf. blarney).
It is therefore likely that animal’s, or person’s, hind leg off is simply a hyperbolic extension of to talk, emphasising the speaker’s persistence or eloquence—a similar phrase is to talk through (the back of) one’s neck, meaning to talk nonsense.
The earliest instance of to talk a horse’s hind leg off, described as an “old vulgar hyperbole” by the English pamphleteer and journalist William Cobbett (1763-1835), is from Cobbett’s Weekly Political Register (London) of Saturday 9th January 1808; he remarked, about a debate that had taken place in the House of Representatives at Washington, D.C., on 27th November:
They all talk; and talk for a long while too. The old vulgar hyperbole of “talking a horse’s hind leg off,” if ever it be verified, will find its verification in the American Congress.
The second-earliest instance of the phrase is also from Cobbett’s Weekly Political Register; on Saturday 17th August 1816, William Cobbett published a letter to Sir Francis Burdett, Bart., in which he wrote:
His Majesty’s Ministers […] do not know what to do with the nation. They are at their wit’s end. They cannot make people give that which they have not to give. Mr. Canning is come home very opportunely to assist his colleagues in making the useful discovery, that though a man may “talk a horse’s hind leg off,” he cannot talk it on again.
The earliest instance of the variant to talk a donkey’s hind leg off that I have found is from The Leeds Times (Leeds, Yorkshire) of Saturday 24th June 1854; it quoted the Weekly Dispatch, which criticised the fact that in the British Parliament “talking men usurp the place of doing men”:
The House swarms with lawyers; and these gabbling geese are scarcely ever off their legs. There is a jobbing Cicero, called Malins, for example, who, to use an expressive phrase, would ‘talk a donkey’s hind leg off.’
The English author Charles Whitehead (1804-62) had used the variant to flatter a donkey’s hind leg off in Richard Savage: A Romance of Real Life, published in Bentley’s Miscellany (New York) of August 1841:
‘That Joe—that Joe’s as deep a put as here and there one. There—he flatters up that fool of a husband of mine, that he makes him believe he’s one of the seven wise men; when, if the truth must be told, he’s no more brains than a broomstick. I wish we could get shut of him; but he’s bound for five long years. That fellow ’ud make a milestone believe that the coach couldn’t run without it, and ’ud flatter a donkey’s hind leg off—he would!’
A pun on an early variant described as “a common saying”, to talk a dog’s hind leg off, appeared in The Era (London) of Sunday 21st June 1846:
Giving Tongue.—There is a common saying that some people will “talk a dog’s hind leg off,” but I know many who would “talk the four (fore) off?”—Lavater Junior.
I have found yet another early variant of the phrase, this time referring not to an animal, but to a person; Bell’s Weekly Messenger (London) for the week ending Saturday 7th December 1839 reported that on Tuesday 3rd a watchman named Charley had appeared at the Middlesex Court of Requests to claim fifteen shillings from a lady named Sergeant:
Mrs. Sergeant’s maid-servant, on behalf of her mistress, positively denied that any agreement to pay a stipulated sum had ever been made. The plaintiff, for his trouble, now and then received a present in money.—Mr. Commissioner Dubois: But you admit your mistress did consent to this man watching the grounds.—Charley: And ring up her maid every morning at six, round the back way, cause she said I might holler and holler my hind leg off, afore I could make her hear in front.