If it's a first generation, I have some tips for you, since I had the same problem. If not, I have no help for you but I'll go ahead and describe what I did anyway in case there are any first generation owners having the problem.

The first thing to check is that you actually are trying to charge it. I had completely forgotten that the first generation Kindle does not charge via the USB cable. It only charges via a small barrel jack. So of course when I plugged it into a USB charger, the charging light did not come on.

The first generation has a removable battery, which looks like this: http://i.imgur.com/5Bv0EA3.jpg

I took it out and checked it out on my multimeter. It was down to 2.3 volts. That's bad, and many chargers will not try to charge a lithium polymer battery that has been discharged that far. I assumed that this was why mine was not charging (I still thought at this point that it was supposed to charge via USB).

I had done some reading on recovering over-discharged lithium polymer batteries. It turns out there is quite a bit on the web about that, largely at sites in the radio controlled model vehicle communities. Apparently, those people have somewhat of a habit of accidentally running their batteries into the ground (all puns intended).

The danger in trying to recharge an over-discharged battery seems to be that the internal resistance goes up the more overly discharged the battery is. Chargers will push current appropriate for a regular battery through them, and the over-discharged battery will heat up much more than a regular battery because of the increased resistance (and also produce gases at a higher rate). The result is that you can cause the battery to burst into flames.

You can actually buy special pouches to put your lithium batteries in while they are charging that are designed to contain lithium battery fires and explosions...the very existence of these is scary [1] [2] [3].

Based on my reading, it seemed that as long as I kept the current low it would be safe to try to recharge the battery. I kludged together something on a breadboard to try this: http://i.imgur.com/UMW7EH2.jpg

The power supply on the write is producing 3.3 volts. Normally you would charge a lithium polymer at 4.3 volts, but my goal wasn't to fully charge the thing. I just wanted to get its voltage up to where the Kindle would be willing to charge it (at this point I still thought it charged via USB and so was assuming the reason it would not charge was the battery being so low that the kindle considered it unsafe to try).

The resistor was chosen so that (3.3v - Vbattery)/R was around 10 mA. The hope was that this was sufficiently below the normal charging current (probably around 150-300 mA) that even if the resistance was high it would not heat the battery too much, and I intended to frequently feel the battery and stop if I detected any heating. (The two leads going out of the picture go to the multimeter, which is monitoring the current).

When I started this going, I got the current I expected, and detected no heating of the battery. As I watched, the current started falling off, and the battery continued showing no signs of heating. I took this as a sign that the battery voltage was actually going up, and so the current was going down. I cut power and measured the battery voltage, and it was indeed up to around 2.5 volts!

I switched out the resistor for one that would give about 50 mA, and started charging again. The battery did not show any signs of heating, and the current again started going down as the voltage came up.

I kept doing this, lowing the resistance (aiming now for around 60-70 mA) and running until the current had fallen significantly, and then repeating, until finally I was using no resistor.

I got the battery up to about 3.2 volts this way, and actually got a some signs of life out of the Kindle. Still would not charge in the Kindle (and I was still thinking it was supposed to charge through USB).

Since it was now well above 3.0 volts, and so in the range that should be acceptable to most chargers, I switched over to a little USB-powered lipo charger I got from Sparkfun: http://i.imgur.com/Rw0Mpzx.jpg

I let that run a while, checking frequently to make sure the battery was not heating. I'd stop on occasion to check the voltage, and it was going up. I let it go up to around 3.7 or 3.8.

The Kindle would now boot and I could connect it to the computer via USB, which means I had a means to install the update without relying on wifi. I didn't want to use wifi yet because I wasn't sure what the battery capacity was. I've read that over-discharging could reduce the capacity of the battery.

I noticed that the charge light did come on when the Kindle was plugged into a computer while the Kindle was running, but I wasn't sure if it was actually charging. This was when I happened to look in the manual for something, and saw that for charging you have to use the barrel jack. I had completely forgotten about that. I then searched a while and found the Kindle power adaptor, plugged it in...and it charged!

So now I'm left now knowing if it would have actually been find back when it was only 2.3 volts, if I had just found the charger and used it, or if it would have refused to charge because the battery was over-dischargred and so my breadboard kludge to inch up above 3.0 volts was necessary.

You may have noticed from the battery photo that it has 5 wires, when a single cell lithium polymer battery only needs 2 wires. Taking one of the black wires as reference, the other black wire and the white wire are at 0 volts, and the two red wires were initially at 2.3 volts, and rose as I charged it. There is no resistance between the two black wires or between the two red wires. There is 11k ohms between the white wire and either black wire.

It is also possible to buy old Kindle batteries at reasonable prices, such as this one for $5.32: http://www.powerbookmedic.com/Amazon-Kindle-Battery-Replacem...

[1] http://www.amazon.com/Dynamite-Li-Po-Charge-Protection-Small...

[2] http://www.amazon.com/gp/product/B00T01LLP8

[3] http://www.amazon.com/Safty-Guard-Charging-Large-180mm/dp/B0...