Hey! When you go through the key you automatically get it(spider room). The door to the right of the sauna opens and inside theres the green key(also disappears when you go through it or just touch it). When you have that too and go back to the swiming pool the last door is the green to try, in front of the sauna thats the exit :)
So basically the key should disappear with a sound effect indicating that you have it on you. (There is no options to take it out, or inventory system)