I had been under the (slightly uneducated) impression the US had sort of island hoped its way across the Pacific, clearing out the Japanese forces bit by bit. So I am curious to know why this strategically well located island was left to the Japanese until their eventual defeat.
Because there were more useful locations that were taken instead.
Wake was most important for keeping the Hawaii Philippines route open, as well as scouting the Marshall Islands to the South. With the fall of the Philippines the Allies shifted the strategic focus South first to defend Australia, and then to the Solomon islands.
However once the counter offensive of the Allies got rolling it bent South of Wake and seized the Marshalls instead. This allowed for the usage of large easily accessed lagoons as fleet bases which was not to be found at Wake.
From there the fleet moved West and North again to take the Marianas. Islands there such as Guam, Peleliu, and Tinian were large enough to support heavy bombers like the B-29 to reach the Home Islands. And to then support the invasion of the Philippines and Home Islands.
Wake was simply left to wither on the vine and bombarded into impotency.