Siliguri (SGUJ) is the nearest stop to Sikkim. 12424/New Delhi - Dibrugarh Town Rajdhani Express is the fastest train to KishanGanj/New Jalpaiguri Junction.
If you are looking for economic one, North East Express or Sikkim Mahananda Express would be better. Note only Sikkim Sikkim Mahananda Express has stoppage at Siliguri. You will also get train connectivity from KishanGanj to Siliguri.