George Canning
For 119 days.
Then he died.
He first entered politics under the mentoring of William Pitt the Younger. Canning became an MP in a rotten borough (a borough with so few voters that it was easy to bribe them all) in Pitt's government.
Canning's bit was a great personal charisma and writing skill. This caused him to skyrocket though the party because, well, they were very short on orators, and he gave them a rhetorical power that they lacked. He also more or less invented public campaigning in the United Kingdom.
As a result of all this, he got himself a circle of followers but also managed to alienate many more, risking losing political allies for personal reasons. He once reduced Lord Liverpool to tears with a long satirical poem mocking Liverpool's attachment to his time as a colonel in the militia. He then forced Liverpool to apologise for being upset.
When Lord Liverpool died, King George IV chose Canning as prime minister, over the heads of The Duke of Wellington and Sir Robert Peel. As such, neither of them was willing to serve under Canning in his cabinet, which would have been as much a problem for him as it had been for Spencer Perceval. Except he died and became the shortest-serving prime minister in British history, although he still managed almost four times longer than William Henry Harrison.