Can we extract KI from Lugol solution?
As you know you have to remove the others species like $\ce{I2}$ and $\ce{I3-}$. $\ce{I3-}$ I'm quite sure it exists only in solution so simply evaporating the water from Lugol 's solution you should eliminate it. To remove completely $\ce{I2}$ you have to do some others steps. 

$\ce{KI}$ is produced industrially by treating iodine with $\ce{KOH}$.(see Wikipedia for reference). 

$$ \ce{3I2 + 6KOH -&gt; 5KI + KIO3 + 3H2O}$$

So I think the simple way is to treat Lugol's solution with $\ce{KOH}$ make an ice bath to precipitate $\ce{KIO3}$, filter the solution to remove it and than evaporate water. 
You can also try to remove iodine by means of active carbon I know this method is used to purify water see ref and ref.
You can try evaporating the Lugol's solution and re-crystalize it and than eventually  try different solvents to solubilize one of the compounds and leave the other as precipitate but if think is quite difficult to find a better solvent than  water (as you know iodine is very insoluble in water instead $\ce{KI}$ is very soluble).

If you want to make the $\ce{KI}$ from your self here there are two Home experience:


extract potassium iodide from salt
make $\ce{KI}$ from $\ce{I2}$ and $\ce{KOH}$