Catalysis of the reaction: ATP + cytidine(34) in tRNA(Ile2) + L-lysine = AMP + diphosphate + H+ + lysidine(34) in tRNA(Ile2). This modification converts both the codon specificity of tRNA(Ile) from AUG to AUA and its amino acid specificity from methionine to isoleucine.
tRNA(Ile)-2-lysyl-cytidine synthase activity