Open the door in the mountain in E7
+3 for getting key, +2 for unlocking mountain door
First you will have to get onto the Gnome's Island. You will get the key if you don't guess the Gnome's name after three tries. You guess his name, talk to him (Gnome is circled in image to left).
The key is an alternative to the magic beans you get if you do guess the Gnome's name. You can still complete the game if you don't guess the Gnome's name first time though, getting the key is probably easier.