K(H) is a simple C^*-algebra. For if I is a nonzero closed bi-ideal of K(H), then it is a closed bi-ideal of B(H), so by [Mur, Th. 2.4.7] F(H) Í I, hence K(H) = [`F(H)] Í [`I] = I. Therefore I = K(H).

Ref.
[Mur] G.J. Murphy, C^*-algebras and operator theory, Academic Press, 1990.