Suggestions for Our "Extra" Day in Anaheim?

(Used to live down there, have been back for Disney & to visit a few times)

1) You'll need a car no matter what

2) Personally, I'd go to the beach.

3) I'd plan for the beach in the middle, but go last day if we were having too much fun. HOWEVER we tend to have better luck doing 'non-Disney' on day ONE before checking into the hotel (or just dropping off luggage). If this were MY trip, we'd fly in, check in & drop off luggage, head to the beach or the pier for the day, come back, start Disney early the next morning...4 days at Disney -we alternate park and hotel pool to keep refreshed-, shuttle, fly out.

4) Avoid Anaheim. It's a hole. (Drugs, prostitution, gangs, etc). Safe enough to drive through, but you don't really want to stop. I'd recommend dropping over to Huntington and heading south to Seal Beach (lemme check my map for a moment, I also lived in San Diego, and occasionally mix up my beach names! ETA: Yep! Totally got that right. Seal Beach is families, seniors, and surfers... LOL wanted to make sure I wasn't sending you to SD or a party beach. I don't even read the signs, just drive on autopilot from my old house) for swimming... Or over to Venice Beach or Santa Monica Pier for the tourist factor. LOVE the pier. Although, I personally live swimming, more.

5) Plan at LEAST 3 hours ahead to get through LAX if that's your airport. Bad traffic can make that a 2 hour drive (so 5 hours means leaving at 1pm), although Sunday should be RELATIVELY calm... I'd want to be shuttle distance away (with guaranteed arrival time... The shuttle should list out shuttle a leaves at x arrives at y, and the transit times will be different for every day and different times of day). Hence not planning a lot for your last day any distance away.

((The parts of LA that are NOT safe to drive through are East LA, and South Central LA... Just as a heads up. Anaheim is FINE to drive through. Just don't stop there))

If you don't have a car you won't be able to get anywhere in LA, public transportation is pretty much non existent.
I love Legoland but that's a good two and a half hour drive each way (and lots of traffic) so it wouldn't be worth it unless you were spending the night.
If you don't want to spend every day at Disney I would suggest renting a car for one day and going out to the Santa Monica Pier, or Venice Beach. I've never been to Knott's Berry Farm but I've heard that's fun for younger kids, and if your kids are a little older they would like Universal Studios :)

4 days in a row of Disney is going to be tiring, but I would not try to do Disney on Sunday when you are scheduled to leave. I'd even get 2 or 3 day passes if you want to see other sights in SoCal. I would personally use Sunday as a day to sleep in, have a nice breakfast maybe at Goofy's kitchen or one of the character theme restaurants and then prepair to head home. Just sounds like too much in too little time.
But if you're really set on it, I'd go somewhere closer on Sunday that would be more easily accessible like the Discovery Science center, Santa Ana zoo, or Long Beach aquarium. Legoland would be way too far. It's going to be expensive though if you plan to use a taxi.

ETA: although I agree with other answers that Anaheim isn't the most beautiful place, not all areas are that horrible. Depending where you go. I've lived in OC all my life and yes once even Anaheim. It also has a newer shopping/entertainment center called Garden walk with some nice restaurants and a theater. But I agree with the others, forgot to mention the beach. I'd do Main st. in Huntington Beach it has nice little shops and restaurants and a Ruby's diner on the pier.

Erika. Southern California has the WORST public transportation of any largely populated area in the U.S. And Anaheim is no place to site see, its just plane booty. Yes, I would break up your week with something else. I would strongly recommend you rent a car for that day or just pay for a taxi if you stay in Orange County. Just forget about the bus system. there is a train that runs up and down the coast that leaves out near Angel Stadium, but it goes to almost no points of interest. So you are still at the mercy of taxis even it you take the train. Theme parks are pretty exhausting, how about a day at the beach to cool off (Anaheim is a little inland and always hotter than the coast). Again, just rent a car or call a taxi.
If you have it in you to do yet another theme park, the San diego Zoo is worth the drive. Lego Land is a big hit with little kids, but I for one can't do 5 days of theme parks and rides.
If it were me I'd want to spend the day in Laguna Beach because its a quaint walkable city and you can go to the beach or stroll the shops and restaurants. In August they have the Sawdust Art festival which is fun.

I'd do something different on Thursday (or Friday) to break up all the Disney. And I'd do Disneyland on Sunday because you will feel less rushed there than if you were going someplace new for just that one day.

LA zoo is not so great. I'd do Legoland. You will have less traffic heading south from Anaheim to Carlsbad than if you try to drive into LA especially on a weekday. It is also cooler in Carlsbad this time of year. It should be easy to rent a car for just one day. Try enterprise or Alamo for a good rate. Often they will pick you up and drop you off from hotel to the car lot.

Another option for a day off from Disney is to hang out at the beach. Doheny and Laguna are fairly close to Anaheim. Laguna has their art festivals going on right now. Doheny/Dana point is further south and less crowded, a good place to go to get away from the heat and crowds at Disneyland.

The beach!!! Such a great time to relax - the polar opposite of the "go, go, go" of Disneyland.