Using the flat sheet of gasketing rubber, you are going to make a replica of the old gasket. If the old gasket is too badly destroyed, you will need to estimate the new gasket's O.D. by measuring where the new gasket will sit.
You can see the three "tabs" in the inner ring of the gasket. The stock gasket has these tabs and they slide into a little lip in the valve to keep it in place. Make this reasonably accurate, but it's not rocket science.
Using the vice and pick again, reinstall the gasket you just made. You will need to do some poking around, but it's pretty easy to get it into place.
Now's a good time to test the valve. Wash off the air pump side of the valve and suck and blow into it. Make sure no air is leaking past the gasket. I also sucked on the vacuum nipple and activated the clamping valve. Make sure it operates and reseals. If not, your valve failure might be an actual failure of the vacuum powered valve itself and not the gasket.
Reintall the old clamping ring and further secure it with a hose clamp. Using a hammer, tap and round the hose clamp band so it cannot slip off.
Reinstall the valve on the car but do not attach the secondary air pump hose or the vacuum line. Start the engine and make sure there is no exhaust air blowing backwards. (A good way to test you exisiting valve!) If the engine was cold, quickly attach the vacuum hose as vacuum pressure should be activate shortly after start on a cold engine. Once the valve opens, you should now have exhaust air blowing back. Wait for the the vaccum air to stop again (or pull the vacuum hose) to make sure the valve closes and seals again properly.
** Note: I'm sure there's a better choice in gasket material out there, but this is a super cheap, easy to obtain solution. **