Ok, since you "inherited" the project, anything is possible at this point.
560ohms is a lot of resistance for a "superbright" LED meant for 5v. So I would venture a guess that these are 12v units, not 5v. Make sure the ones which lit at 5v don't have a different resistor. As you can see, it's not rocket science to replace the resistor so they work at 5v.
As for why some lit after bypassing the resistor, that's anyone's guess. From the looks of the poor solder job, I would guess a cold solder joint. If you have a multi-meter, check for continuity with the solder joint between the probes. Then measure the resistance of the resistor. If you do both of those things, you will be pretty certain where the problem lies.
If you don't have a meter, get one. It's the only real way to troubleshoot this kind of thing.
RandyT