Hi Scott,
I use Dirtex by Savogran to wash any piece of furniture I am working on. It's close to sudless and removes dirt and grease from the finish or what's left of the finish. And that is the point, finding what shape the finish is in. The finish is the protection/window to what is below it.
Using 0000 steel wool, ring the wool out and wipe in the direction of the grain. Work on areas the size in which you can immediately wipe clean with paper towels and wipe dry.
This particular piano your talking about sounds incredibly dirty. I wouldn't even suggest to the customer taking this on.
Cleaning a piano this way is a lot of work.
This best left to a professional furniture restorer.
Unrelated tip for using this washing method:
I use this method to clean key frames when rebuilding grand actions. Because the frames are usually unfinished, after wiping away the loose dirt with paper towels I also hit the area with a hair drier.
When I first tried this I thought the water/Dirtex solution would soak into the unfinished oak frames. But drying it with the hair dryer revealed very little penetration into the cells of the wood.