Unfortunately, Nidoking doesn't learn Horn Drill at all. In order to have your Nidoking learn Horn Drill, you'll need to level up your Nidorino to level 46 (you learn Horn Drill at that level) and then evolve it into a Nidoking.
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No, you need to have your Nidorino learn it at level 53. that's why i saved my moonstone until level 53 and then made him a NidoKing.
Lots. Earthquake, horn drill, mega kick etc...
Nidoking doesn't learn Horn Drill by level. Its previous evolution, Nidorino, does, which learns it at level 46. You'll have to level up a Nidorino until it learns Horn Drill, then evolve it to Nidoking.
Look bub, i think you might teach it a TM i don't know, or it just might learn it by a level.
Yes, Nidoking is capable of learning Surf (HM03).