Other LIs are Burushaski, Okinawan, Ainu(extinct?), Sumerian and Pictish(both extinct). There is some evidence that Japanese and Korean are related because of similar syntax structure. For this reason I would not include them on the list of LIs. Feel free to add to the list.
Okinawan is universally accepted as the closest living relative of Japanese. The family to which both of these belong to is known as "Japonic".
Ainu is not extinct, but there's evidence that it is also related to Japonic and Korean within a larger Altaic family.
The genetic affiliations of Pictish have not been demonstrated to everyone's satisfaction, but there's a strong case for it being an Indo-European language of the Celtic family.