Well first thing is sleep in eventually have something to eat and then relax around the pool for a good amount of time with some drinks. Later its resort hopping and either Downtown Disney or the Boardwalk for some shopping, dinner and finally some party time at a bar or two.
Other options include a long day trip to the Space Coast for some beach time, shopping and relaxing.
So many choices!: sleep in, relax, go for a swim, mini golf, gym, resort hop, downtown Disney, Disney quest, boardwalk area, watch a movie by the pool at one of the resorts that does this, go to the chip n dale sing a long at fort wilderness. Obviously i don't do all of that in one day, or one trip
I usually don't take a day off from the parks but I do leave one day for rest by taking the morning and early afternoon off. Since you are staying at AKL, I would recommend reserving a table at Boma for breakfast. Take advantage of your hotel and eat at their restaurants
We slept in, enjoyed coffee on the balcony, laid out by the pool, grabbed lunch by the pool, kids did all the afternoon pool games and activities, explored the resort (so many nooks and cranny's). Had a nap, headed to DTD for dinner at Wolfgang's and pin trading. Came back and got in the hot tub and enjoyed a few Safari Amber's while the kids played and checked out the Animals thru the night vision goggles. I think that's it!
I don't do "non-park days." With the relatively low cost of adding additional park days to a multi-day ticket, it seems wasteful to me. Rather than go 100% theme park commando every day and then taking a day or two off, we generally take it easy in at least one way each day we're there. So instead of a full "day of nothing," we might sleep in one day, take a nap another day, go to bed early one day, etc.
Head over to Fort Wilderness and explore. There is a petting zoo, horseback riding and a couple of shops. You can rent bikes (which are fun to explore with and a road that takes you all the way to Wilderness Lodge) You can look for the lawnmower tree, there is a hiking tail along Bay Lake that we have seen lots of deer on. You can rent a canoe or rent a fishing pole and fish.
What I do is stuff at downtown Disney, look at other Disney hotels, and play miniature golf at Fantasia Gardens in terms of stuff on WDW property. I also go to a Monorail hotel at night to watch Disney's Electrical Water pageant. What I do at Downtown disney is look at shops, have a sundae at Ghiradelli's, watch Cirque DU Soleil La Nouba at night.
I done Busch Gardens and Sea World in the past when it is a non WDW park day.
-Relax at the resort
-Explore AKL and other deluxe resorts aka resort hopping
-Go to Downtown Disney (shopping, dining etc)
-On property mini golfing
-Water parks
-Have a nice sit down breakfast possibly (Ohana's is a great option)
if you plan on leaving the resort I can't help you there, I stay in my bubble
Fort Wilderness has Pony Rides, Horseback Riding, Carriage Rides, Fishing and Archery.
Rent a boat and tour the Sassalouga (rent from DTD).
Fishing or a Carraige Ride at PORS.
Don't usually take a full day off, normally. But a lot of times I'll sleep in, have an afternoon at the pool with a few drinks, go DTD for dinner etc maybe a spin on the highway in the sky go check out some resorts etc that or even though I guess it is a "park" but I'll go to Blizzard Beach
As we are there for 14 days (rope drop each day) we take one off day each of the two weeks. Then we usually go to Downtown Disney for some good shopping, lunch etc. We've been to Tampa Hard Rock Casino, St. Petersburg beaches, visit family that live about 40 minutes from WDW and sometimes just roam around the area stopping at various places and maybe trying some place new for dinner.