I am not sure if the Kinect was plugged in when your dog chewed your cable, but the board could have been shorted while he/she was enjoying the meal. If you are sure the board is not shorted and splicing the cable still is not working, I would recommend buying a broken kinect on ebay and swapping the cable, and the bonus is that it does not require you to know how to solder. If the cable still does not fix it then you know the situation is much more dire.
Lazy Links:
[http://www.ebay.com/itm/As-is-Genuine-Microsoft-kinect-sensor-bar-/271152274734?pt=Video_Games_Accessories&hash=item3f21ef5d2e|Broken Kinect]
[http://www.ebay.com/itm/Kinect-Sensor-Broken-As-Is-/281052894980?pt=Video_Games_Accessories&hash=item41700ed704|Another Broken Kinect]
[guide|4066|Kinect Teardown] (So you kin* find out how to open it up.)
''*Please don't mind the puns.''