You can redeem when you have all 4 items in a line highlighted. if it's still gray, you cannot redeem. The inventory is the list at the bottom of the screen when you go into design mode. The redeemed items will be in there- for instance, the utensil rack is in the wall item tab, etc.
There currently is a problem with the chef hats moving- many people have complained that the hats move but go right back to 0 when you exit out and come back (myself included). it's just a bug that'll need to be worked out.
Hope this helps! If you have more questions, please ask!!