Brute-force is not very Elfish, so i'll go with a more clever and nimble solution: Energy efficiency.
If our elf uses as many calories per km as a human sprinting, she will overheat and die very quickly.
If she has speed and caloric mileage comparable to a human marathon runner, she will face the same thermal challenges, which means she'll have to wear very light clothes, leave the mithril mail at home, and require frequent hydration, including salts and electrolytes. This is independent of her endurance, stamina, etc. The volume of water a person can sweat before dropping isn't that much.
So, higher metabolic rate isn't the answer. Also, as explained by James, higher power output creates more waste the body has to get rid of.
She will need to do it with less energy that a human, which boils down to: how to make running more efficient?
When we humans run, most of the energy is consumed in raising the center of mass of the body once per step, and this energy is not recovered when the center of mass is lowered again. Very little energy is expended towards overcoming air friction.
This is why someone in decent shape can ride a bicycle at 20km/h with less than 100 heart rate. This is a barely noticeable effort which can be sustained almost all day. Can't be compared AT ALL to running at the same speed...
Running at 20km/h should require 2-300 watts of mechanical power, which due to our efficiency of around 25% translates to about 800-1200W input chemical power (ie, food)... and output thermal power (ie, heating).
Now, let's draw inspiration from an amazing animal:
As red kangaroos hop faster over level ground, their rate of oxygen consumption (indicating metabolic energy consumption) remains nearly the same. This phenomenon has been attributed to exceptional elastic energy storage and recovery via long compliant tendons in the legs.
So, kangaroo have KERS (Kinetic Energy Recovery Systems) like F1 cars.
Let's equip our Elves with a few elastic tendons in their legs. Now, picture an elf running. When the front foot touches down, the knee and ankle joints flex, and the springy tendons store energy, which is then recovered to propel the elf forward on the next stride. Just like kangaroos.
Springy tendons also reduce the strain on joints and muscles, which is important for long distances.
Plus, this should give our elf a nimble, agile and quiet gait, which is very in-character.
Since Elves make the best swords and the best gear, they also make the best shoes, so she'll have blister-proof Fantasy Air sneakers.
Let's also give her an elvin agility bonus, and fudge this to 50% energy savings.
We're down to 100-150 watts, which is still a bit less efficient than a mountain bike with big knobbly tires, but a lot better than before.
Now, we give another elvin bonuses:
- Increase metabolic efficiency to 35% instead of the human 25%
- Optimize metabolism to burn fat more efficiently than humans.
She still produces 100-150 watts mechanical. However, increased efficiency means she needs to burn less calories to do so, therefore sweating a lot less, and needing less water. She doesn't need unrealistic lungs or heart either. She will also burn mostly fat. So, she needs 40 grams of fat per hour, or 30 grams fat plus 20 grams sugar. Pastries (ie, Lembas) should be fine.
Reducing (not increasing) her metabolic needs seems to be the way to go.
Fat will carry her much further without eating, as body fat reserves are quite important (even for someone who looks like a stick figure) whereas sugar reserves are not.
Now, some of that energy could also come from a magical source, of course. Or her magic could help with the pesky logistic details, like joint pains, blisters, etc.
Just don't make her run for a week in plate armor.
Elves are more efficient, thus use less power, and as a result Legolas arrives at the end of the trip with fabulous hair, looking relaxed, and without breaking a sweat.
Meanwhile, the dwarf sweats gallons in his heavy armor, stinks, and blames it all on the elf.