I've been fighting with this silly "kobo will only charge if I hold the cable at a specific angle" issue for... oh, probably a year now.
I finally got fed up with it, decided to open it up and see what parts I'd need to order to have someone solder a new USB port on.
I found some instructions for opening it up (use a plastic spudger tool -- one of these:
https://www.ifixit.com/Store/Tools/Spudger/IF145-002-2 -- to unsnap all the clips around the outside, pull the bezel off and make sure the tape stays attached to the screen, unscrew the 4 screws in the corners) and when I opened it up, I found that the solder joins were nice and solid... the problem was simply that the metal surround for the plug had bent open.
a couple of minutes with a pair of needle nose pliers and a tiny screwdriver, and it's a tight fit again, and my kobo is happily charging up to full without any wrapping the cord around it and hoping it stays at exactly the right angle.
Sorry I didn't take any photos, but it was really super easy to fix, so I went ahead and did it.
If anyone else is suffering from the same issue -- give it a shot.
(Just don't squeeze it *too* tight - that's what I needed the tiny screwdriver for, to pry the edges back open a bit)
I'll probably have to do it again eventually since it's already been bent, it's likely to be weaker... but I think I'm going to switch to a 180 degree microUSB cable, which should cut down on the strain.