Here's what I want: A mod that makes it where, when you go under about 20% stamina, spells fail more often, jump height is reduced, and power attacks aren't available. And where attacking with weapons, running, jumping, and aiming a bow all drain stamina.

The problem is... I've found myself with absolutely no mods I can find that make stamina important. Most of the stamina mods are about increasing carry weight, or making stamina regenerate faster. Also, while Requiem does exactly what I want with stamina, it causes so many changes, it's not compatible with other mods I use. I've also seen a few running drains stamina mods, but they weren't what I was looking for.

Does such a mod even exist?

Just a stamina mod, I doubt it.

SkyRe uses Stamina to calculate damage dealt and blocked. These are complete overhauls and will change the whole game.

Deadly Combat (end of time as developer moved on?), it makes attacks and blocks use 'more' stamina.

SPERG has this and a few other interesting things without changing too much gameplay:

Ethereal Arrows - Firing a bow without any arrows equipped will fire an ethereal arrow from your magicka pool. These arrows improve slightly with perks that affect bound weapons.

If you run out of magicka, you run out of arrows (or you use real ones...). I play it and never ran out of arrows, yet

I also like the overheal perk, which gives you temporary extra hitpoints.

I use the http://www.nexusmods.com/skyrim/mods/1931/?, jogging and running mod. This gives me a lot of immersion over sprinting everywhere and extreme-sprint only costs stamina. The jogging is slightly faster walking and you do not need to change any gameplay, by making jogging stamina dependent.to compensate, I use Better Vampires and can travel at ~200% speed at night which is probably the normal jogging speed of vanilla game (during the day I am like a match waiting to be fired up and slow).