some more things to check
try the 'working' laptop's power brick on the supposedly faulty one just to rule out any potential issue with the power brick itself. We had an ASUS F3Sr power brick start failing, the screen would flicker (old style backlight not LED), it would trickle charge the battery over a day instead of a couple of hours or not at all if the laptop was turned on. A cheap Chinese clone part from Amazon has worked fine for years since.
if the battery is really charging properly in the other notebook (going to 100% and then lasting the expected amount of time when in use), check the contacts in the potentially faulty one are clean and not misaligned.
LIFEBOOK A512 (FJNBB29) - Intel Core i3-3110M, 16GB Crucial Micron DDR3 1600 RAM kit, 525GB Crucial MX300 SSD