Then I reassembled the top half of the headset and tightened the locknut, again using the mop handle to keep the fork from spinning:
The important thing is to check for play in the fork after tightening the locknut. If you can feel the fork rocking back and forth in the frame, then loosen the locknut and screw the bearing race in until play disappears, then retighten the locknut. Check the fork again to make sure it can turn easily.
Next I greased the front wheel bearings and reassembled the cones:
Now I wanted to work with the bike upside down, so I reinstalled the seatpost and handlebars:
Next I greased the bottom bracket bearings and slid the cranks into the bottom bracket shell:
The hard part is trying to maneuver the cranks through the shell, so don't push the bearing retainers into the cups until the cranks are through in order to give yourself enough clearance. Once it's through you can tighten up the bearing cone until play disappears:
Now replace the keyed washer and tighten up the locknut. Spin the cranks and see if they turn easily without rocking back and forth. If there is play, or the cranks don't turn smoothly then loosen the locknut and turn the cone to solve the problem (tightening the locknut can make the cranks feel harder to turn so adjust for that).
The spokes needed a lot more tension before I would consider the wheels safe to ride on:
Tensioning and truing wheels takes a little practice, but here's some more info on it. I also took the front tire off to patch the inner tube.
After bending the left pedal back into shape and dripping oil inside both of them, I could reinstall the pedals:
Make them nice and tight, and put some oil or grease on the pedal threads first.
The brake levers were a bit too low on the handlebars, so I decided to unwrap the tape and reposition them:
There should be a screw inside that's visible when you pull the lever down, you can use a flat bladed screwdriver to tighten or loosen it in order to move the levers.
Now for my least favorite part:
I only hate doing cables because I always get my hands covered in oil, and used cables don't like to slide into the housings at first since the ends are flattened out by the pinch bolts. I always drip oil into the housings first, then coat the inner wire in oil before threading it through to reduce friction (this makes a big difference to how your brakes and shifters feel when you're done). Once the cable is threaded all the way through the housing you can pull it tight and secure it by tightening the pinch bolt:
The front derailleur's housing was too long, so here is how I measured the correct length:
Remember my broken saddle? Well the rails were still intact, and I just happened to have a nicer Schwinn seat with a broken rail. I drilled out the old rivets, then cut and bent the rails to fit the Schwinn saddle:
Now the bike is mostly finished:
All that's left to do is ride it around for a while and adjust the brakes and shifters as needed. It should be a pretty solid bike now, and it didn't cost much money. It only needed some time and a little bit of skill to fix it up.
Edit: The brakes would squeal, and I noticed that the pads were angled away from the rim. I used an adjustable wrench to turn the pads the other way, giving them an inward toe. This immediately quieted the brakes:
You should really only try this on cheap steel brakes. Heat treated aluminum calipers might crack so don't even think about twisting those. Again, be careful and don't overdo this (and if your brakes don't squeal then don't worry about it). You might also have to center the pads around the rim, so loosen the nut behind the part of the frame that the brake attaches to. Then rotate the brake until it's in a good position and retighten the nut. If the whole brake rotates as you tighten the nut, try oiling the threads before screwing the nut back on. That's about all I have left to say. Finally, check your shifters and make sure the limit screws are set properly on the derailleurs. Cheers.