Without knowing the exact dimensions of that cabinet, I know I have had very good luck with mounting 21" VGA monitors in Taito cabinets, using the existing monitor mount. You probably will want to take the case off (should be 4 screws, and then jimmy the cap off the tilt base at the bottom, there will be a screw under it.
The monitor then sits right on the mount. You could strap it in if you wanted additional rigidity, but since it is such a low angle (about 45 degrees) it sits on there just fine for me. Would only have to take it out if I moved the cabinet a great deal.
You could also custom make this with some wood and L brackets. Simply build the enclosure where the monitor would site out of wood, and slide that around your monitor and slide the monitor and mount into place. You can then bolt it in.
Easier said than done, but once you take the case off, you will see the tabs and edges that could hold it into place.