I think your way is a definite improvement on the original, which has some problems:
- A double negative: "couldn't" with "neither ... nor"
- A scope problem for "with", which does not apply to "his claws" the way it is written, although it needs to.
If you still want to use "neither ... nor" and also address these two issues, you could write "The eagle could attack the serpent neither with his sharp beak nor with his claws".
But both that and your version allow the possibility that the eagle could attack in another manner again, let's say with his wings. If you mean he had no way of attacking, and the claws and the beak were particular ways that were excluded, then you need a "non-restrictive" modifier, which calls for a comma and putting the modifier affirmatively: "The eagle could not attack the serpent, either with his sharp beak or with his claws".
(By the way, I'll just mention that there's one other minor issue, which is the ambiguity in the antecedent of "his". As all of these examples are written, it might be taken to refer to the serpent rather than the eagle. But we'll let that one pass, and say we should know from the context somehow.)