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The battery charge controller chip is underneath that piece of steel shielding/heat sink. Removal probably requires a hot air gun made for surface mount soldering (800 degrees F.). You could try to get it off with an iron and desoldering wick or solder suction tool, but there’s a significant risk of damaging the diode on the battery connector side or the resistor on the opposite side.
I have the same problem with one of these. However, I noticed that the battery connector was not fully seated (it took a magnifying glass to spot it), so when I the charger is available again (it’s charging another Kindle at this time), I’ll try it again. If it still gets hot and won’t charge, the charge controller chip is most likely bad, and I’ll have to pull the shield or send this one to the recycler.
All of the black wires on the battery are connected together, and all of the red wires are connected together. However, charging a lithium battery is not so simple as connecting a power source and waiting a while. Excessive charging current or voltage can cause a nasty fire or explosion. The white, yellow and blue wires to the battery are for temperature sensor and possibly other monitoring circuitry. or sensors. It is not a good idea to bypass these.
If you are intent on manually charging the battery (for example, if it is discharged below the safety cutoff threshold and you want to try to salvage it, limit charge current to 0.08 Amps (80 mA), and absolutely do not let the voltage applied to the battery exceed 4.20 Volts. Watch the battery to make sure it does not overheat or swell; if it does, disconnect power immediately.
It’s much safer and easier to replace the broken USB jack and let the on-board power management chip take care of charge control.
Watch for variants at Step 6 and 8. The iPod I'm working on has the copper tape on the rear-facing camera wrapped around the side of the camera. A tab on the steel shield makes contact with the tape, but the tape is not stuck to the shield. In this case, do not remove the copper tape, just leave it alone. I still had to lift the camera along with the logic board to get at the digitizer cable.