Answer to: After change audio jack the home botton dont workMore then likely you damaged the logic board while soldering in a new audio jack. Logic boards are very easily damaged with heat and having not opened one up myself i'm not positive on the schematics or layout of the Ipod Touch 2G. Check your logic board for cold solder joints and any padding lifted up on the board.
Answer to: How does one distinguish a dead battery from a dead logic board?Well the obvious simple answer is if the ipod touch responds to having the cable plugged into a usb port obviously it's not the logic board if it completely boots up. Where as What ABCellars said is also correct.