A cougar hung out around here last year at this time. it killed 5 goats a mile from us. That lady had donkeys for guardians. The donkeys don't have to be faster than the cougar, just faster than the goats. Donkeys are prey animals.
The cougar was on the property right next to us one night screaming, the dogs were going nuts, but the cougar didn't come calling. My sheep were safe. The cougar was heard around the corner from us, close to the back of our property. It moved on to easier meals.
disclaimer: This is based on my own research and limited experience with llamas and donkeys. I own sheep, and one llama and am planning to buy a dog in the near future. Cougars are not a problem where I am but I have several friends who’ve delt with them for years.
Dogs are generally accepted as the only certain protection from bigs cats, cougars are pretty shy and opportunists, they don’t like the big scary barking, dogs are carnivores, and most cougars are barely bigger than big LG dogs anyways. Donkeys and llamas can be effective against dogs and coyotes, but you need to make sure you have the right kind, what I mean is their mentality/personality. You want a confident one not one that’s shy or uncertain. A donkey or llama will protect sheep simply because it doesn’t like canines and it sees the sheep as it’s herd (only works if only one donkey or llama, not multiple). Donkeys can be very effective against all these predators if you have one that really likes the sheep and is very protective, same with the llama, for these guys tho it comes down to personality and maturity, and they are much less effective with cougars unfortunately.
So in summary, yes donkeys can work but if cougars are a problem for you I’d invest in a dog, it does take longer unless you can find an adult for sale but you’ll have much better peace of mind IMO.
last thought, whichever LGD you decide on (you can have multiple as long as you only have one donkey or llama each, dogs will work together but the llamas and donkeys have a tendency to form their own herd if there are multiple llamas or mult donkeys) spend some time reading about the introduction and training process. Livestock guarding is generally innate in the good LGs but it is possible to be done incorrectly. There are several Facebook pages besides the www and this forum full of information. Hope this helps.