So the dog chewed through the control wires from the heating unit to the condenser outside. I have replaced those wires. BUT now the board on the heating unit in the garage seems to not have power. I am guessing this because the board usually has a red light on when it's plugged in, but it's not on anymore.

Picture of the board is below.
http://identium.net/ebay/board-27.jpg

I am sure you have blown the fuse or burned the transformer up. Is the transformer just below the board in the picture? I can't see anything that looks like a fuse inline or on the board. Part of the board is hidden by the wires. Check to see you have power to the transformer, and back out and to the board. If the transformer is bad and it didn't have a fuse, I would add one using an inline fuse holder.

I hope you are doing something to keep the dog away from the wires. My one puppy not only chewed the control wires, but the foam over the vapor tube, plus was chewing the flex on the 240 supply. I replaced the flex with rigid conduit and the plastic disconnect with a metal one. I also put metal stove pipe over the control wires and foam.



http://www.photolocker.net/images/Labman/KatetheAC.jpg