Where can I find this part: l3p08x-56g20 ? Thank you

Your battery has been depleted past a chargable state.  This is a due to a bad design in the phone that doesn't monitor the battery power state well enough.  It actually allows the battery to be taken under the minimum amount needed to recharge the battery. 

You can fix this, but do so at your own risk.

You can simply purchase a new battery, that will work, just been sure not to let it deplete the battery totally anymore.

Now, on to the fix

-Take out the battery.
-Get any usb cable you can find sitting around. (any usb wire will do, an old printer cable, anything with a usb end that can plug into your computer)
-cut one end of the plug off. You need to leave the side on that connects to your computer. Cut off the side that plugs into a device (printer, camera, web cam, etc)
-Strip the rubber off the wire and expose the wires inside.
-Separate the wires from each other. You will only be using the BLACK and RED wires.
-strip the very end of the black and red wires carefully using a wire cutter exposing the metal wire inside the red and black covering.
-Plug in your home made battery charger to your computer. Careful not to touch any exposed metal, as it will give you a small shock.
****WARNING****
DO NOT UNDER ANY CIRCUMSTANCES ALLOW THE RED AND BLACK WIRE TO TOUCH EACH OTHER
****WARNING****

-now carefully hold the black wire to the negative terminal on the battery (it should be noted with a black circle with a - in the middle). Then touch the red wire to the positive terminal (it will be noted with a red circle with a + in the middle).

Hold the 2 wires there for 20 seconds. Then quickly plug the battery back into the phone, and then plug it into a regular charger. Give it little bit of time, if you've done everything correctly the charge light should come on.

I had to repeat the last step again to get the light to turn on indicating it was charging.

You must not hold the wires to the battery for too long, 20 seconds should do it. You can overload a battery with this trick and actually make it explode if you hold it there too long. You are well within the few minutes needed to detonate the battery due to overcharging by sticking very strictly to 20 seconds.

Just be careful and you'll be fine.

One you get a charge light give it a few minutes in the off position, then try powering up the device and keep it plugged in until you get 100% charge.

Good luck,

Peter