Spring!
Time for allergies and mosquito bites and repairing all of the yard tools you broke in the Fall. Time to make the yard look pretty so you can drive around snobbishly saying, “pfft, our yard looks way better than their yard.”
The first thing I needed to fix was the mower. I have a sturdy old John Deere (JA-62 if anyone cares) that has cut down more than its fair share of evil grass blades over the years. I’ve changed its oil, air filter, sharpened and re-positioned its blade. When being lazy last year and mowing up leaves instead of raking them, I accidentally tore through the ten year old nylon pull cord.
But hey, it’s just a pull cord, easy fix, right?
No, wrong. It probably should be an easy fix, but when I’m involved, things gets stupid complicated stupid fast.
The culprit:

It went from fine to frayed to filament to fuuuuuu, in record time.
Step 1: Gather your tools
You will need:
-A Phillips head screwdriver (or drill with Phillips head bit)
-A flat head screwdriver (or same as above, with a different bit, obviously)
-Sockets of various sizes (depending on your make/model of modest mower)
-A socket extension or socket screwdriver attachment (to leverage or not to leverage, that is the boring question)
-Beer (your choice)

Since it was slightly overcast today, I chose Smuttynose IPA.
Step 2: Remove the shroud and gas tank
To get at the housing for the pull cord, you need to take the mower completely to pieces. If you find yourself removing big, tight bolts and looking into the crankcase, you’ve gone way, way too far.
Remove the shroud (aka pointless plastic mower-helmet). Mine was secured with two longish Phillips head screws.
Once the shroud has been removed, you’ll need to remove the gas tank. This is a great time to spill gasoline all over your shoes and jeans, if you haven’t done so already.
This is also a good time to note what other parts probably definitely need replacing, so you won’t be shocked when they break. I noticed how ragged and leaky my gas supply hose was, and I took a picture of it so you can commiserate with my future repair job.

There was a chick on TLC's "My Strange Addiction" who drank gasoline. Beer is cheaper.
Step 3: Remove the weird thingy that contains the wound up pull cord
There should be a few bolts holding the pulley/flywheel/wind up thingy in place. Check to make sure you aren’t just removing screws at random; I accidentally removed the throttle cable and it was a total bitch to put back on without a lot of swearing and kicking.
One weird thing to note with this piece: you’ll also remove the oil dip stick when you pull it loose. From my research, this is normal for all types of mowers, so don’t judge your mower too harshly. It’s what nature intended.
Once removed, flip this piece over to find the pulley that hides the cord. If you’ve made it this far in under an hour without calling anyone for help, you win the “better than the writer of this article” award.

Turbines to speed!
Step 4: Remove old pull cord
Using your flat head screw driver, remove the screw that holds the pulley in place.
Spoiler alert: this thing is under tension and will jump out at you like one of the “can-o-snakes” from the 90s. Set your beer down before proceeding.
If your pulley housing looks like this, you’ve either done everything perfectly, or screwed it all up:

Surprise! You get to spend more money than you thought!
Ignore the mess you’ve made, and continue to remove the old cord. If the pulley spring is getting in your way, just detach the little hooks from either end and set it aside for now.
Most mowers are surprisingly simple. The pull cord is probably just knotted on both ends. Cut the cord using a sweet, sharp knife, and set it in the corner of shame (the trash can).

While less popular than "Rock, Paper, Scissors", "Knife, Rope, Pulley" found some use in the small engine/farmer's market crowd.
I know this step is getting kind of long, so you’ve probably wandered off. Stay with me! I promise rewards!
Next, remove and cut the cord from the handle, making space for your new cord. You’ll probably have to stick something (a screwdriver, twig, lightsaber themed chopstick) into the handle to get the knot dislodged.

I have no idea why my pinky decided to go all "fancy-pants holding a martini glass" while taking this picture. Feel free to make fun.
Step 5: Attach the new cord and re-coil your pulley
Because trying to order directly from the John Deere website was like chiseling my request into limestone and sending it via barefoot courier, I went to Lowes and bought a universal mower pull cord. It wasn’t quite as nice as my original cord, but beggars and choosers and all that.
Note: The Lowes staff does not appreciate a random, dirty dude walking around their store taking pictures of their products.
Push the new string through the hole in the pulley and tie a knot. I have a tendency to tighten knots using my teeth. Don’t do that in this situation, unless you like the taste of dirt, gasoline, nylon, and used motor oil.
Now comes the most excitingly annoying part of the whole process: rewinding the pulley. You have to take the part that sprang out at you and carefully coil it all back into the center of the pulley. Fair warning: this make take multiple attempts.
Ultimately, you want it to look like this:

Much harder to do that it looks, if it looks simple.
Step 6: Replace and wind up the pulley
Now that the new cord has a new home, you can start to rebuild your mower. The pulley mechanism has two small plastic “wings” that act as a stopper and cause the wheel to wind back up once it’s pulled. Be sure to line these up correctly, otherwise your cord won’t rewind at all and will hang limply, like a very sad, very dead fish.
Winding the cord up is relatively simple. Before you tighten down the flat head screw, turn the entire pulley clockwise, pulling more cord into the center of the device. If you are doing it correctly, the “wings” will resist and pop out, spinning the entire contraption backwards a little bit.
You need to wind it up more than you would think. Don’t leave enough cord to reach all the way to handle holder, as this won’t create enough tension to pull the cord back into the pulley.

These are the aforementioned "wings." They work sort of like a yo-yo. But not really.
Step 7: Test
Replace all the parts you took off, and give the pull cord a yank. If the mower starts AND the cord retracts fully, you win!
If not, you lose. Repeat Step 6 until you have wound the cord to the sufficient tension.
Step 8: Mow you magnificent bastard, mow
Victory! You saved your mower’s life. You are a hero to your backyard. I’m sure the hydrangeas will throw you a parade.

At this point, dance a celebratory dance. Or not. Your call.