Here's what I posted awhile ago if you forget to click the link.
Go to the junk yard get a bulb connector socket cut it off so you have the wire leads. Get a connector socket such as the ones for the side marker and cluster signals bulbs. Polarg 194/168 (M4) Hyper White Wedge (pair) those are in the EC.com store they don't cost to much. These tend to last just as long as your dome light bulbs. You can also get anything that would be compatible to fit the area where you need lighting for so you can easily fit it in the area without cutting, drilling, dremeling out. The most you may have to do is dremel out an area if your bulb connector is extremely huge. I'd reccomend you buy something that has inexpensive bulbs that any auto supply store would carry.
Find a good socket size so you can tap into the area's that needs lighting.
You don't need a new board for the most part just make sure it is funcational if the lighting is the problem here's what you do.
1. Use this new bulb connector socket. With Pos.(red) & Neg.(black) connections so you don't accidently pop the bulbs when wiring them up.
2. Get a long wire to run through the fire wall.
3. With the wires now inside the cabin run them behind your damaged board.
4. Find an area where you can place the bulbs in the board and mounted where it lights up the most you can use mulitple connector sockets to tap areas that need more lighting.
5. The cig. ring lighter, a.c. & defroster buttom take apart find a place to luminate area you may have to run another bulb or a simple soldered l.e.d. connection their as well radioshack sells some l.e.d's already soldered for you with the pos./neg. wire lead look around trust me you'll find something even a computer l.e.d.'s have wires as well. You can check their too just for more sources.
6. Last you want it to dimm with the rest of the lighting so having it connected to your low beams you will be able to control it from the dimmer area unless you want an external dimmer. Unless you plan on a steady power feed you can tap it into your factory dimmer as well. If your not to good with electrical just stick with the low beams for now.
This should help you get sort of the idea hope you can work with it. For #5 If you can't light up these areas and you find it a struggle but you have fix the power to light up some of it more power to you and I would call it a job well done. But if you do have patience and light up the rest of it without the board then I call it an EXCELLENT JOB WELL DONE!
One one more thing the A/C control unit board should be sold by Toyota as a whole unit. The unit only has 2 connector plugs in the back one is small and the other is larger in size both are the color white. These connectors power the whole A/C & Heater controll unit.
You may be seeing the unit from the top view I have to ask you first when you say "looks like the board is connected to the system via this 8 or 9 pin white band of wires that is soldered to the board." When you ask this question have you already taken apart the whole unit or just the circuit board?
The connector does have a band of wires that's white in color you can see this from the top view but it also has a removable connector at the end that it's inserted to the back of the unit for easy installation just like the speedometer cluster one of Toyota's easy access wiring techniques. All you do is plug the two connectors into the back of the A/C unit. Ask the dealer if the $60.00 is the whole unit I think it should be the amount of the whole unit with board. It's just plain rediculous that Toyota would sell that circuit board separately not very practical. I'll try to post up some pics of what it looks like later I've done it quite some times when I took apart the whole dash and messed around with it here and their it's a very simple process.