Root cause analysis
If the phone had charging problems before you bought the new cable and still has charging problems after you tried the new cable then the problem is likely to be the phone and not the cable.
Check that the battery is actually dead. Use a multi-meter or find someone that has tools. If not then do not proceed with Plan A or B.
1a. Search the web to see if there is an easy way to power the phone other than through the USB.
(a) Maybe through the battery connectors with an AC-DC power block? I'm imagining paperclips + phone + DC block, one clip on inside of tube, one clip on outside of tube, ...add tape. Make sure DC voltage matches the battery spec.
(b) Maybe with some of the manufacturing test connectors?
2a. Get the device charged up.
3a. Get all of your data out of the phone.
4a. Wipe the phone. Use recovery to wipe as much as possible. Figure out how to completely wipe it and return it to stock. Make sure its stock everything: kernel, recovery, ROM. Format the SD card.
5a. Let battery discharge *almost* completely. Just enough to demonstrate that it works.
6a. Return it.
If that fails, plan B:
1b. Try to disassemble the phone (tools needed)
2b. Find USB wiring specification for which pins are power
3b. Inspect those pins for problems
4b. Solder to fix problems (you may also be able to wiggle/hold the connector "just right" to make contact which saves you steps 1b-5b)
6b. Go to step 2a at top, at your discretion of course.
If Plan 0, Plan A, and Plan B fail then re-post.