What piano piece is Emil Nekola skating to in his short program?
I found an answer after some additional guesswork and checking with some music recordings.

The piano music is an excerpt from Modest Mussorgsky's Pictures at an Exhibition, specifically "The Hut on Hen's Legs" section. The soundtrack before the episode cuts away to the end of Emil's performance clearly matches this; the piano music starts where this section of Pictures at an Exhibition begins and follows it (before the cut) without any alterations.