In episode 13, after Kousei played Love's Sorrow, he played this one song on the piano. What is the name of that song? I've been looking for it for the past four hours, but haven't been able to find it.
I'm assuming you're referring to the piece Kousei plays at the very end of the episode; this is the only instance where Kousei plays the piano in ep. 13 after his "Liebesleid" performance. I recognised this as a Chopin composition, possibly a nocturne.
After some searching, I found that this was Nocturne no. 2 in D-flat minor, op. 27. (Score, recording by Eric Lu at the 2015 Chopin Competition.) It should be relatively easy to recognise the piece if you have some prior familiarity with it, since Kousei starts playing at the very beginning.
Originally, I wondered if some other piece (such as "Liebesfreud", which is named in another answer as the piece Kousei plays) might have crept into the soundtrack at some point, perhaps in a flashback or in Kousei's reflections. However, after listening carefully to the soundtrack and checking the "Liebesleid" score and some recordings, I've confirmed that everything we hear during the performance segment is from Rachmaninoff's "Liebesleid". Thus, the only music for piano that we hear being played in this episode is "Liebesleid" and the Chopin nocturne.
Note that what viewers hear is a modified, lengthened version of the Rachmaninoff arrangment. (The soundtrack returns to the beginning of the piece right before the first key change, when we see Kousei underwater. This jump does not exist in the original score. There are similar jumps later, during the flashback with Saki after Kousei has lashed out at her, and when Saki reflects on Kousei.)
That is Sergei Rachmaninoff - Liebesfreud(Kreisler, arr. Rachmaninov).