The Japanese did successfully invade the Aleutian Islands. One theory is that Japan lost the Battle of Midway because they were sending troops and supplies to Alaska.
The Americans in the end didn't even drive them out. The Japanese decided that it wasn't worth the time, supplies, or manpower in the end and snuck who was left on islands by submarine back to Japan.
The last couple of islands that the Americans landed on to take back from Japan were actually abandoned by the Japanese and American forces then died from friendly fire due to the dense fog.