Dissociation of ligand
Three different types of dissociation can occur here, the two mentioned by the OP:

$$\ce{K[Ag(CN)2] &lt;=&gt; K+ + Ag+ +2CN-}$$
$$\ce{K[Ag(CN)2] &lt;=&gt; K+ + Ag(CN)2^-}$$

And one where only one cyanide ligand dissociates from the silver:

$$\ce{K[Ag(CN)2] &lt;=&gt; K+ + AgCN + CN-}$$

However, as Klaus points out in his comment dissociation $\ne$ ionization. Ionization is a chemical reaction that generates ions where there were none before. Consider the ionization reaction of ammonia $\ce{NH3}$ in water:

$$\ce{NH3 + H2O &lt;=&gt; NH4+ + OH-}$$

I'm not sure if this type of reaction is possible, although the cyanide anion is the conjugate base of a weak acid $(\ce{HCN}, \ \text{pK}_a =9.21)$. Thus, any liberated cyanide ion could react with water (assuming this is an aqueous solution!), though I would hardly call this an ionization (as there is no net increase in ions).

$$\ce{CN- + H2O &lt;=&gt; HCN + OH-}$$