The halogen bulbs are 12 volts at 20 watts. Almost certainly the "High" setting is 12 VAC, and the "Low" setting is the same 12 VAC but through a half wave rectifier (just a diode). So, if you want to use the current switch and processor board, you'd need something that could be driven by 12VAC, both with and without a half wave rectifier. I'm guessing this would be tough to find; the best you could probably do would be a system that works when the switch is on "High", and doesn't work, but also doesn't burn up, when the switch is on "Low".

The halogen bulbs are 12 volts at 20 watts. Almost certainly the "High" setting is 12 VAC, and the "Low" setting is the same 12 VAC but through a half wave rectifier (just a diode). So, if you want to use the current switch and processor board, you'd need something that could be driven by 12VAC, both with and without a half wave rectifier.

The problem is that half-wave-rectified AC is a lot different from pure AC; it has a large DC component to it. Some LED drivers could handle it, others couldn't. I doubt you'll find a manufacturer that will sell their product as compatible with this, as it's too rare an application for them to worry about. So, you can try various products, but you may find they won't work, and you may even find that the "Low" setting damages the bulb/circuit.

An alternative would be to install your own switch, but getting it to look good would be tough.