There will be a full bar and meal service after take off, which will take at least 2 hours, then "breakfast" will start around 90 mins before landing. Actual flight time is actually around 5h45m. If you take the meal times off that (when the lights are on full, and you can't really recline) you're left with around 2 hours of cabin darkness. Which ever way you swing it, it's going to be tiring, and you won't get a good nights sleep.

Personally I'd try and wake up early that day, eat before the flight, have a drink only to try and send me to sleep, then don an eyemask and ear plugs and try and sleep til landing. As to travelling with family, I guess that means kids, and I don't know what would work best for them.

Breakfast will certainly be worth skipping though - try and get an extra bit of sleep instead, and get a proper breakfast in London.