There may be easier ways to do it but could you dissolve the remainder in water and ppt the chloride (from the KCl) with AgNO3 and determine grams Cl^-, then compare to the the theoretical amount of KCl produced if all of the KClO3 were decomposed to KCl.
KClO3 decomposes to KCl and O2 which it escapes. A sample of KClO3 that has had some decomposition will contain some KCl.
A sample of this mixture could be dissolved in water and tested with AgNO3.
Pure KClO3 would not produce a precipitate since AgClO3 is soluble in water. If some KCl is present, AgCl will precipitate.
I disagree with this answer.:-) Everything you say is true about the KClO3/KCl; however, the question isn't asked that way. The question is, "How can we tall if all of the KClO3 has decomposed?" That means we have already driven the oxygen away and we are left with mostly KCl. Of course we will get a reaction with AgNO3. The question is how do we determine if there is a molecule or two of KClO3 that did not decompose and we must do so with a bunch of KCl mixed in. The procedure outlined will work to distinguish KClO3 before it is heated but not after it is heated. And before it is heated, we already know the answer.