Hi,
I am thinking your faulty board is dead.
I have run the accel demo on my boards just to re-acquaint myself with them.
With the one that works, I presume if you power up with no switch depressed then no LED's. Yes?
If you power it up with a button pressed then LED 4 flashes quite fast indicating it is transmitting. Yes?
With the "dead" one powering up with button pressed or not all four LED's come on straight away and stay on. Yes?
The programme never turns on LED's 1 and 2 so this is why I think its dead.
If you are not sure how to programme/debug them then try some measurements with a multimeter. Battery disconnected and Serial cable disconnected.
If you look at the PCB with the buttons at the bottom the right hand side of all the LED's should be connected to 3V. Measure this to the left pin of J103 (marked +)
The left hand side of all the LED's should measure 3.5Megohms to 0V on J103 (marked with a - vertically) and to +.
Now connect the battery and test for 3.0 volts at J103 when turned on.
Leave it on for 5 minutes then turn it off and then touch all the chips with your finger, none of them should feel warm or hot at all.
Let us know what you find.
Regards David