I found 18 songs titles with 16 of them playable from YouTube videos here on a site called The Cavan Project http://www.thecavanproject.com/18-songs-about-january/

1. “January” by Pilot 2. “January Rain” by Hunters & Collectors 3.“January On Lake Street” by Atmosphere 4. “January” by 28 Days 5. “January Song” by Billy Bragg 6. “January” by Disclosure 7(unavailable). “January 28th” by J. Cole 8. “January Rain” by David Gray 9. “January Hymn” by The Decemberists 10. “January 28th, 1986” by Owl City 11. “January” by Wildfront 12. “You In January” by The Wonder Years 13. “January White” by Sleeping At Last14. “Last January” by The Twlight Sad 15. “June In January” by Dean Martin 16(unavailable). “January” by Louden Swain 17. “January Sun” by Fizzy Blood 18. “January Man” by Christy Moore