I hope you get good sleep along the way coz this might take long
There are direct flights and trains for this journey. New Delhi (Delhi) to Bhubaneshwar (Orissa) distance is 1742 km and you can get there by flight in just over 1 hour and 55 minutes or by train fares starting from ₹ 547.
The heat there will make you sweat, but it's surely bearable.