Well-I think you mean L.A. and not LA (from one and lived a long time in the other) because of the Legoland mention. We never made it to Legoland but I heard it was very fun. We did Disneyland at least once a year while we lived there because it was so close and well, Disney is just awesome.
Pasadena is GREAT for kids...there is the Rose Bowl which is cool to drive over to see and they can run around if they want as people are always exercising around the perimeter....all free. Right near the Rose Bowl is a Children’s Museum...not really an L.A. only thing but still fun and I don’t think it is too expensive.
Drive down Colorado Blvd (the shopping district area) and Huntington Ave in Pasadena as both are really neat and quite lovely. There is a very fun store to take them to called STATS to just walk around. Sounds crazy but we would take our babies there a few times every year to see and sometimes shop-is a decoration store and they get all decked out for each holiday, Halloween is probably on tap end of Aug and every time we go back to visit, I take my kids there now.
Hollywood area Grumman’s Chinese Theater is so fun...walking around the front area outside to look at the handprints from the stars and some of the sidewalk stars-all free of course is very L.A. and you can make it very fun for 4 year olds by talking about the action movies they like or whatever...optimal for pictures too.
One idea we loved was taking the kids to the Getty....I am pretty sure it is free and there are kid friendly parts to every section etc...what makes this so L.A. is the tram ride through the Santa Monica Mountains and the lovely grounds too which get you up close and personal with the wonderful L.A. mountain perspective.
Another great area is the Santa Monica Pier....walking on the pier, then driving down farther North to Malibu or South to access the beach 10 or so blocks South of the Pier is very fun. Don’t advise swimming in the ocean around there off the top of my head due to water quality and stay away from the beach access right near Santa Monica or the Pier area, lots or homeless and scary looking folks there.
Also very fun is the Manhattan Beach area- nice little downtown area you can park pretty close to the beach and walk to beach to see ocean and the stores around there are nice so you could get an ice-cream if it’s hot.
All in all, if you are not from the West Coast being on the beach in any of these areas at sundown (except the Santa Monica Pier area I'd not advise being at too late) is a very fun event for little ones and is a great time to explain the natural phenomena up close and live of the sunrise-sunset as the sun seems to drop slowly into the ocean.
A couple of dining recs if you want some fun L.A. kid friendly experiences-
Islands is a chain and has locations all over the greater L.A. area....good burgers and great fries and a surfing feel. They are kid friendly and feature closed circuit TV’s throughout all their restaurants which run loops of way cool footage of people surfing and doing great water sports which is a great thing for them to watch while food is coming, etc.
Also good is Marie Callendars which has lots of good kid friendly things and great pie desserts (especially their fresh strawberry pie which should be in season).
One other place which is a bit more $ but not outrageous is in Malibu (you can go to the beach there before or after you eat) Gladstone’s 4 Fish....ideal location, fun atmosphere with peanuts on the ground while you wait and they have beach view seating and very fun "I love L.A." sing along that happens every day at a certain time (not sure when but if you call them)...all 3 places are very adult fun too.
Oh, did I mention that I love L.A....have a great trip and enjoy the way hip town and the wonderful people who live there.