Catalysis of the phosphorylation of tyrosine and threonine residues in a c-Jun NH2-terminal kinase (JNK), a member of a subgroup of mitogen-activated protein kinases (MAPKs), which signal in response to cytokines and exposure to environmental stress. JUN kinase kinase (JNKK) is a dual-specificity protein kinase kinase and requires activation by a serine/threonine kinase JUN kinase kinase kinase.