You mention it's only one part of your island. Would it be possible to move your mob related activities over to a diffrent part of your island?
Two reasons you are lacking in the mob department. 1. Mobs like the dark, your mob trap has light leaking in from the opening in the top 2. That magnum torch stops mobs from spawning in a healthy radius. 128 blocks horizontally and 32 vertically. Your Mob farm is 20 blocks vertical and 40 blocks from the torch meaning no mobs would be able to spawn.
You also need to ensure that you are far enough away from the spawner as mobs are very shy about spawning around players.
You also need to ensure that you are far enough away from the spawner as mobs are very shy about spawning around players.