Approximate amenability of K(X)K(X)

We show that if a Banach space X   is “fairly close” to a Hilbert space, then the Banach algebra K(X)K(X) of all compact operators on X is approximately amenable. We use this to give examples of Banach spaces X   for which K(X)K(X) is approximately amenable but not amenable. Thus we answer a question on existence of such spaces, open since 2000.