PSP charges battery but won't turn on

My PSP battery charges up just fine, in fact it can still hold that charge (I used a four-volt led).

But, the PSP won't turn on after I turned the Power switch up.

I tried using two other batteries and they both seemed to work just fine so it's not the PSP.

did you allow the LED to discharge the battery completely?

Yes it discharged the battery completely, but when I plug it in the PSP it would recharge fine without starting.

If other batteries work fine, just assume that particular battery is defective/broken, and replace it.