Where do I even begin to clean this chassis?

[captainmoody]

Spray that chassis down with an industrial cleaner-degreaser (take it easy around the transformers)then hose off with warm water. As long as it dries off it will be fine.
The place where I used to work did that with industrial monitors before repairing them, they had to as they would be full of grease and coolant from cnc machines.
Never had any problems and the units looked like new when finished.
In the end it is someones own personal preference as to how they want to do it, personally I have repaired hundreds of monitors and televisions over the years and always preferred the above mentioned way.