Personally the story dragged for me, but overall interesting and emotional. The "500 miles" which is a 1960's revival song - it took me back to when I was enlisted in the military and shipped overseas - to Japan of all places- For that year I held the record in long distance phone bill LOL God I missed home! But anyway I think they only used a portion of the song in the drama.

So here is the answer to your question as to who sing the song:

500 Miles- Justin Timberlake (Inside Llewyn Davis)