Lithium-Ion batteries don't retain their charge well over time - it's no surprise that the Pismo can't boot from it. As for the power adapter, the original kind had a tendency to break at the junction between the thin cable and the plug into the computer. These tended to flex a lot and the wires would separate, making a nice little spark gap. Mine made an acrid plastic burning smell.
This doesn't seem tob be your problem, but try replacing that cabe anyway. If you take the thing into an an Apple Store, and the Genius is agreeable, you can get it fogired out whether it's the power adapter or something inside the computer (which is doubtful if you didn't bump it severely).