A very long time.
A query at the Phrases and Sayings Discussion Forum asked if the British slang term for 'a very long time' was donkey's years or donkey's ears. My first thoughts were, "donkey's years of course - what would ears have to do with it?". It turns out that I was being rather hasty. Donkey's years is now the more commonly used slang term when meaning 'a long time', but donkey's ears, although used little in recent years, has been a jokey alternative for some time - certainly from the early 20th century, viz. E. V. Lucas' Vermilion Box, 1916:
"Now for my first bath for what the men call 'Donkey's ears', meaning years and years."
This slightly pre-dates the earliest printed version that I can find of donkey's years, in the US newspaper The Bridgeport Telegram, 1923:
"With a heavy make-up, you'll be the cutest vamp I've seen in donkey's years."
It is quite likely that donkey's ears was the earlier form and that it originated as rhyming slang, in an allusion to the length of the animal's ears. Donkey's ears/years is often shortened just to donkeys. That is characteristic of rhyming slang, as in syrup (of figs) - wig or plates (of meat) - feet.
Donkey's ears works as rhyming slang whereas donkey's years doesn't. In rhyming slang the last word of a short phrase is rhymed with the word that gives the slang meaning; for example, trouble and strife - wife, apples and pears - stairs, etc. It makes little sense for the phrase to have originated in slang form as donkey's years, as that would rhyme 'years' with 'years'.
The migration from donkey's ears to donkey's years was no doubt aided by the belief that donkeys live a long time. There's some truth in that. Lively Laddie, a donkey who had lived up to his name for many years while plying his trade on Blackpool Pleasure Beach was, until his death at age 62, a contender for the 'oldest living donkey' title.