There is no direct connection from Darbhanga to Darjeeling. However, you can take the train to New Jalpaiguri (NJP) then take the taxi to Darjeeling. Alternatively, you can travel from Darbhanga to Darjeeling in around 5 h 6 min.
From Darbhanga Junction (DBG) train to New Jalpaiguri (NJP) then taxi to Darjeeling. 13 h 49 min ₹2,499. From Darbhanga taxi to Darjeeling. 5 h 6 min ₹6,314. From Darbhanga drive to Darjeeling. 5 h 6 min ₹2,725. From Darbhanga drive to Janakpur airport, fly to Bhadrapur airport, then drive to Darjeeling. 22 h 36 min ₹9,372.