Would ionic compounds dissolve in methanol?
You don't need to do compare the said enthalpies. Maybe the process of dissolution of $\ce{KI}$ in $\ce{CH3OH}$ is energetically favorable, maybe it isn't (the latter seems more likely to me); that's irrelevant. Thanks to entropy, everything dissolves in everything, if only in minuscule amounts.

You just need to compare solubilities of $\ce{KI}$ in methanol and hexane, which basically boils down to the comparison of solvation enthalpy in methanol vs. that in hexane, and here the difference in their polarity provides a clear and obvious answer.