This problem is normally worded with a frog that can summon power for a 3 foot leap once per hour, during which he slips back 2 feet. However, this version of the problem is not worded to suggest there's instant addition of 3 feet every hour. Thus, the assumption is that when he gets to 27 feet, he can climb 3 feet and be out, in reality, climbing progress and slipping rates will be somewhat steady.
The other assumption is that Watson is zero feet tall.
Assume Watson is 5.5 feet tall, with a 2.5 foot reach, and lets give him a conservative vertical leap 1.5 feet (these numbers based on my height, my ability to grab the top of a door frame, and some Googling... average male vertical leap is 50cm).
Watson wants the best possible start, so he leaps into the air, grabbing the rope at his apex. His hands are now 9.5 feet in the air, and once they reach the ledge, he should be able to grab and mantle out.
His net progress is 1 foot per hour. Since he only has 20.5 feet to go, he should be out in that many hours.
Of course, if we knew his height, reach, and vertical leap, we could find the exact number of hours as such:
(depth of well - (height + reach + leap)) / (climb rate - slip rate)