How much ft-lb on impact do you need to kill a coyote or a fox?

You need 24 or less

Where did you get the figure of 24lbs more or less as a Coyote caliber?
The 17mach 2 is a short range caliber for small game and varmints. Its fine for its intended purpose but its irresponsible to use it for game that it was never intended for. For an inexperienced shooter such as you shots need to be taken with care as not to let the animal suffer.
From Chuck Hawks site
Small game (CXP1)
The smallest animals, like tree squirrels and cottontail rabbits. are best taken with rifles chambered for the rimfire .17 M2 and .22 Long Rifle cartridges. The more powerful .17 HMR and .22 WMR are fine for head shots, but are generally too destructive with body shots. The .22 LR is about an 85 yard small game cartridge, most suitable for small animals not averaging more than about 10 pounds in body weight. The .17 M2 pushes that range to about 100 yards.
The .22 Magnum is about a 125 yard small game cartridge, and the .17 HMR does for the same size animals out to about 165 yards. A 40 grain .22 WMR bullet has a sectional density (SD) of .114. To insure quick kills, always select expanding bullets for either of the .22 caliber cartridges. It is best to avoid frangible bullets intended for shooting varmints, however, as they are too destructive for use on small game intended as table fare. A fine example of a small game hunting rifle is the lever action Marlin Model 39 with a 4x scope, in .22 LR.
Varmints (CXP1)
The smallest animals typically hunted with centerfire rifles are varmints. Varmints are pests such as rats, sand/sage rats, prairie dogs, ground squirrels, jack rabbits and marmots that typically weigh up to around 15 pounds. This is where the rimfire magnums and the popular centerfire .22's shine. The lighter bullets in the high intensity .24 and .25 calibers can also be useful for shooting varmints, although their recoil and muzzle blast are unacceptable in some situations.
Among the available choices for medium to long range varmint hunting are the .17 HMR, .17 Rem., .204 Ruger, .22 Hornet, .218 Bee, .221 Fireball, .222 Remington, .223 Rem., 5.6x50, .224 Wby., .22-250 Rem., .220 Swift, 5.6x57, .243 Win., 6mm Rem., .250 Sav. and .257 Roberts.
For short range varmint hunting calibers such as the .22 LR (with hyper-velocity loads), .22 WMR, .25-20, .256 Win. Mag., .25-35, .30 Carbine and .32-20 are satisfactory with suitable bullets.
Be sure to select frangible varmint bullets, not bullets intended for small game, big game or military use, when varmint hunting. This is a matter of safety (to minimize ricochets) and also insures quick kills. An example of an outstanding varmint rifle is the Savage Model 12 Varminter Low Profile in .204, .223, or .22-250, ideally with about a 6-18x scope.
Small predators (CXP1)
Small predators such as foxes, coyotes and bobcats are larger and tougher than the varmints and small game animals that constitute their prey. They may weigh up to about 45 pounds and are best taken with any of the centerfire varmint calibers listed above. Within their respective MPBRs the following cartridges should be acceptable with expanding bullets: .204 Ruger, .22 Hornet, .218 Bee, .221 Fireball, .222 Remington, .223 Rem., 5.6x50, .224 Wby., .22-250 Rem., .220 Swift, 5.6x57, .243 Win., 6mm Rem., .25-20, .256 Win. Mag., .25-35, .250 Savage, .257 Roberts, .30 Carbine, .32 H&R Magnum, .32-20 and .357 Magnum. Javelina, although not predators, are in the same general weight class and are commonly hunted with the same calibers from the .223 Rem. on up.
Practically any centerfire .22, .24 or .25 caliber hunting rifle makes a satisfactory small predator gun and some manufacturers offer special predator control models. The Savage Predator Hunter is one such. A fixed 6x or a 3-9x variable power scope is all that is required.

foot pounds never did crap , but a bullet stout enough to penetrate the brain or other vital organ does.

I was going to be smart-aleck and quote the average foot pounds of the .50 Caliber BMG cartridge (13,500 ft lb) but instead I figured I'd simply mention that Dad killed a lot of coyotes with a Winchester Model 94 30-30, and I don't think he thought about it much past that.

Just an observation, but it seems that you are trying to get validation from the members of the forum to get a 17 mach 2 for the purpose of using it for coyotes and foxes. Is this a gun you really want to get? If so, go ahead and do it, but from what I have read in these threads, the general consensus seems to lean toward finding something bigger that is probably better suited for the task. When we still owned our farm, Dad usually had his Winchester model 94 30-30 nearby, and I know he killed a lot of coyotes with that - from my memories of his recounting those shots, it would seem that the 30-30 did a pretty fine job taking down coyotes.

Of course we're talking apples and oranges here - there is a world of difference between the 30-30 and the 17 mach 2, but there is no doubt the 30-30 got the job done. Overkill? Maybe, but it's better to get a quick, sure kill than to wound with a bullet that isn't quite up to the task.

As a side note, he also had a Marlin .44 Magnum lever action that got some use as his standby gun and it likely took out a coyote or two as well.

Foxes go down quick with a LR HP, I've had good luck with Yellow Jackets and Hi Speed HP's from Remington and Winchester.

But Coyotes can be as much as three times the size of a fox, and I think a .22Mag would be a good minimum for most shooters.

"If", the question is "if" a Specific round has the energy. Well it depends on the distance, mass wieght, and bc. Lots of Whitetail deer were taken w/ a .22mag in the 1930's, but the question is how many got away injured? A .22lr would be a better choice at 25yards over a .17H2. I bought a .17HMR and own a .22 WMR that I use for Coyotes. Either shotgun or Rimfire are allowed around here. So Shot placement is key. Ammo selection is also key. For the .17HMR a 20gr Speer out to 100 yards works well, not perfect, but well. For a .22mag, a CCI Maxi HP mag does the trick out to 100 yards. Head shots are best at 100yds, body shots at 50yrds. 17gr .17 are pretty useless around here. Jim

As fun as it is to agonize over how many foot lbs. we need, I think reality indicates that shot placement and bullet construction are more important. Then we get into the debate of bullet construction! and off we go again.
When I was a kid back in the '70s I had the good fortune of living in a remote area rich in furbearers, so I ran a fairly impressive trapline. This gave opportunity to dispatch a decent number of critters with the .22 pistol I carried. When up close to a trapped coyote, or whatever, I had luxury of precise shot placement and the .22short was always effective. I often made heart/lung shots instead of head shots for taxidermy purposes. I also shot truckloads (literally) of white tailed jackrabbits for bait with .22lr rifle. The huge rabbits (hares) were harder to bring down at longer range, say 100 yards, so shot placement was crucial. Only other rifle I had at that time was a .270 so no experience with rimfire mag. from that era, but good lessons were learned.
I make occasional trip to a big game bird farm to control preditors and also black tail jackrabbits (hares). We try to thin the hares because they distract clients bird dogs. The hares are very thick and we bring many guns to try. Most shots are 100 to 300 yds. and they're usually running and very wary. There is a notable difference between leathality between the 22LR and the magnum rimfires on the hares. This applies doubly on the larger preditors, maybe even more. I still believe shot placement is supreme, but on those occassional (almost frequent) instances of marginal shot placement the magnum rimfires show a real advantage in my observations. We also use centerfires and they are decisive with any reasonable bullet.
Now for the surprize... of the many guns we use on the gamebird farm the single weapon that consistently guarantees I will nail more vermin than my buddies is... a lowly open sighted SKS with a trigger job. It has a magical combo of power, accuracy, capacity, and reliability. Ammo is affordable too. We've used all manner of varmint and sniping rifles, assault weapons, shotguns, Olympic Anschutz, big game rifles, etc. Up to 250 yards the SKS is our go-to gun, but beyond that it's gotta be a scoped varmint rig. It's funny and deflating to see a $200 SKS kill more varmints than an expensive rig. I realize this can't apply to all hunting settings, but it sure does in this particular situation I enjoy. Back to the question.. I don't recommend the 17mach2 unless you're after small rodents.