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LiI has a covalent character because Li ion has high polarizing power due to its small size. Therefore LiI is soluble in alcohol. On the other  hand KI is predominantly an ionic compound, therefore is insoluble in ethanol.
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LiI has considerable covalent character (given the small size of the Li+ ion it will cause considerable polarisation in the (large) iodide ion). (Electronegativity values suggest it's about 57% covalent.)
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