Well in your new cabinet there isn't even a chassis hooked up to the existing monitor, it cannot work without one.
Most monitors are going to just want AC power from a main line, and won't require a switching power supply. The only real concern most of the time is if they require an isolation transformer (this protects the rest of the cab from the high voltages created by the monitor chassis). Since you don't even have a chassis in your new cabinet, you can buy a 'universal' chassis that won't require one.
Either that or just replace the monitor outright with a TV or LCD monitor. Personally I prefer arcade monitors, but this is purely a matter of taste and convenience.