My power cord only charges at a certain angle

Could be the cord itself is the problem. I've seen cords that only provide power when they're bent in the right position, probably from a break of the wires inside, but still stays touching enough sometimes. Especially if the laptop and the plug part of the wire plugged into the laptop stay completely still.

Could try testing the cord with a multimeter at the end that plugs into your laptop, and try moving/bending around the cord (slowly if your multimeter takes a second to read).

This good looking page shows how to test the cord & jack too, with nice images like this one showing the cord being tested:

If the voltage drops to zero in some cord positions, and it's definitely a good connection between the multimeter and the end plug, then it looks like a bad/broken cord.
Cutting off the "bad" part of the cord, and soldering it back together could work, don't mix up the +/- wires though.

If not the cord, it could be the jack on the laptop as the rest of the good looking page (& other answer) suggests; says if it's a solder-on jack then it's probably bad, and to just replace the solder-on jack, with the warning Disclaimer: I’ve made these instructions only for people experienced with soldering and repairing computers. If you don’t feel comfortable doing this job, please do not open the laptop or you can permanently damage your computer. Take your laptop to a professional repair shop instead. Use this repair guide at your own risk.

Or maybe some part of the computer board around the jack is bad, or other computer board part. On a related note, I'll bet that most repair shops would charge enough trying and repair your old laptop that it would only cost "a little more" to buy a new laptop; that seems to be a common motivation to buy a new laptop.