Some trains go directly from Copenhagen airport (Kastrup) to Nässjö. Just have a look at www.sj.se
and search for for connections between "Köbenhavns Lufthavn/Kastrup" and "Nässjö".
Try to book as early as possible since ticket prices increase with increasing popularity.
Try both with and without the promotion code "O-Ringen" in the promotion field, sometimes it (strangely enough) is cheaper without the promotion.