Well I applaud you for wanting to expand your palate. You certainly went "all in" with a bottle of Laphroaig Qtr Cask. Let it rest for a few days and go back to give it another go. I used to dislike peated whisky, now I really enjoy it. Keep it in the back of the cabinet for a cold rainy night.
Beyond the obvious peat and iodine, see what else you can dig out of there. I find it quite sweet and malty underneath, and of course I love the briney sea notes.
There's a lot going on in there in the background.