The plastic knob is probably reamed out and isn't grabbing the valve stem. I'd try to remove the plastic knob to expose the stem and see if you can operate the valve with pliers, vice grips, wrench, etc. Then you'll know if you need an entirely new valve or just a stem key.
Take care not to fuck up the stem by turning it with a loose fitting wrench and shearing off the corners. If it's fighting you, don't force it or you could create a much bigger problem.
Don't continue to force the plastic cap off to one side with shims in an effort to get purchase. This will stress the stem and packings and may cause a leak.
I think Diablo Blanco is a heating/plumbing guy so maybe you two can get together and talk about hairy ass cracks.