Since $\ell_\infty(\mathcal k)$ is a dual space, a result by Rosenthal implies that if $X^*$ contains $c_0(\Gamma)$ then it contains $\ell_\infty(\Gamma)$. 

See Corollary 1.5 in [the paper][1] H.P. Rosenthal, On relatively disjoint families of measures, with some applications to Banach space theory. Studia Math. 37 (1970), 13-36. (Correction, ibid., 311-313.) 

 
  [1]: http://matwbn.icm.edu.pl/ksiazki/sm/sm37/sm3712.pdf