My daughter placed a Glow Stick (phenolic oxalate ester and H2O2) on
her Electronic Keyboard and over a few days there was a splatter area
up to 6 to 8 inches away from the glow stick with a substance
(resembles clear solidified syrup, slightly sticky and very hard to
remove.  The plastic under the stick was melted (on piano's bookrack),
and it was very clear the solution did not 'run' but exploded over the
keys.  My question is how do I remove this substance from the keys?

One guess is that the oxalate polymerized. This would be easy to
remove with an acid or base catalyst. The problem is that this is also
the basic structure of many plastics, so you want to use enough acid
or base to catalyze the breakdown of the oxalate (assuming this is
what happened) without ruining the piano keys (you probably knew
that). So start with a weak acid like vinegar and then go to a spray
cleaner with HCl, then if none of that works maybe try that
goo-be-gone stuff that you find at walmart, etc. (which is actually
not an acid or base, but a volatile carbon based compound, pretty much
like gasoline). My worry with that last option though is that it will
most likely ruin your keys. Also, do not use acetone or nail-polish
remover-- this can damage any plastic (as I sadly learned in Ochem
lab).