Yours looks a lot like mine, except for the wire going from the power connector, somewhere under the battery. Where does it go?
It appears that the wire is just being used because the battery solder tabs on the battery don't line up with the PCB hole on the bottom.
The wire from the solder tab on the power connector (AC adapter -) runs directly to the - of the battery.
Also, it looks like the right two connectors on the power receiver are connected by something, what is it?
The right two solder connections of the power connector are connected to the outer barrel of the ac adapter connector. (-)
The left two solder connections are connected to the inner barrel of the ac adapter charge connector. (+)
Not sure where that trace goes. You'll have to look closer and look at the back of the board.
I haven't spent the time to track down all the traces of the power & charging circuitry to work up the charge circuit schematic.
Mine measures 4.25v at the battery when the power is on, so I don't think insufficient charging voltage is the problem. I remember the original recall right after I bought mine, saying that the batteries were defective. At the time, I didn't care, as it never left its cradle, so I never sent it in. But now, it doesn't even work properly in the cradle, and acts like it's running out of power all the time (i.e. light flashes off & on, and the display eventually turns off, only to be revived by pushing the button). I ordered a new battery for it, and will replace that, but if there's something wrong with the charging circuit that I can fix with a simple jumper wire, I'd like to do that 
.
My battery also reads 4.25v when the adapter is attached. (Raw voltage of the "7.5v" adapter is around 9v.)
I think the biggest issue is that the LIR2032 is inadequate for this purpose and there is no power switch.
The LIR2032 capacity is quite small particularly considering the backlight current. A LIR2450 would have been a better choice or even better a LiPo battery that has plenty of power.
And then with no power switch, the battery will potentially sit there with a draw on it while not in the cradle, like when new sitting in the box.
It is possible the f/w is trying to do something smart and turns off everything once a particular battery voltage threshold is reached to try to reduce the battery drain.
And maybe the battery simple can't provide enough current once the battery has a bit of wear on it so the voltage rapidly craters to the point where everything stops working.
No way of really knowing without doing some power analysis or unless TED provides some information about it.
--- bill
