If you're flying BA you'll find the back of seat tv's great. My ds plugged into the cartoon channel and that was that.
If you have a night flight be sure to go to the galley to get snacks as they are usually available there but they don't wheel the trolley up and down while people are sleeping.
Also suggest you take an empty drinks bottle and top it up when the trolley comes round so you can give dc a drink whenever they want it without having to wait on the stewards.
LAX airport is rubbish. Hardly anything to do there at all. Do you have a long wait? If at all possible I would recommend stopping for a couple of days and going to Disneyland on your way back.
Are you going to be there furing their winter or summer?
North island is brilliant. Lots to do. I recommend the kiwi friuit farm at Te Puke here www.kiwi360.com/ Don't be fooled into thinking it will be a sedate golf kart type ride, that thing goes mega fast and ds (aged 5 at the time) loved it better than the roller coasters at disnelyland! You can spend a whole day there - play area, giant pond, kart/farm trip etc. Ds still recalls our visit there (the giant apple slide)
Another good day out is the luge at Rotorua www.skylineskyrides.co.nz/rotorua/
Don't let your youngest one go down on her own though, let her ride with you. Ds persuaded me to let him go on his own the third time and he couldn't work the break and vanished off at speed then crashed! Your 8yo will be fine though.
Agrodome is another good place to visit. Do the sheep show but I wouldn't pay extra for the farm visit again www.agrodome.co.nz/?Add=1&IDAdd=1061307592
www.rainbowsprings.co.nz/page/5-Home another great day out. The fish go mad for the food and you may get splashed!
There's loads of other things, I can't remember them all but we did lots including a Mauri village visit, steam train trip, bird park... Omakoroa beach is pretty and has a park near it if you're in that area.
I enjoyed this more than ds did (he was 3 on that occassion though) but it is a nice place to go and the hot mud and geysers are interesting if you like that kind of thing www.tepuia.com/
Do head over to the bay of plenty, lake tapu, rotorua etc. The tourist information places have loads of info and are really helpful. Only problem you'll have is squashing everything in! Hope you're going for long enough...