I am not sure about the exact surcharge amount. The base fare from Kurihashi (where the Tobu section starts) to Tobu-Nikko is 940 Yen. The part that I am not sure about is the limited express surcharge for that section. The unreserved surcharge is around 1,500 Yen so depending on the distance, the total surcharge is most likely somewhere around 1,600 Yen ballpark.
However, there is really no reason to take the direct trains. They aren't any faster than take Shinkansen. From Shibuya, you can just hop onto Shonan-Shinjuku Line to Omiya to catch Shinkansen to Utsunomiya, and then transfer to Nikko Line to Nikko. Your entire fare would be covered, and you won't get there any slower than using Tobu trains going through Shinjuku.