fred, the "not charging" can be caused by multiple things, like the cable, dock connector bad battery, bad charging coil as well as other components. Check for corrosion, loose connection etc. on the logic board. I marked the most common issues on the diagram. Check for continuity etc.
Start by cleaning the dock connector properly, replace the battery, make sure the battery connectors are properly attached to the board. Worst case scenario, you have issues with the power management IC. check this video for a reflow and the image to see if the jumpers might help you out. Place jumpers across the fuses
. Hope this helps, good luck.