We prove that the K-groups of the Banach algebra script B sign(script J sign_p) of bounded, linear operators on the pth James space script J sign_p, where 1 < p < ∞, are given by K₀(script B sign(script J sign_p)) ≅ ℤ and K₁(script B sign(script J sign_p)) = {0}. Moreover, for each Banach space script X sign and each non-zero, closed ideal script I sign in script B sign(script X sign) contained in the ideal of inessential operators, we show that K₀(script I sign) ≅ ℤ and K₁(script I sign) = {0}. This enables us to calculate the K-groups of script B sign(script X sign) for each Banach space script X sign which is a direct sum of finitely many James spaces and ℓ_p-spaces.