Hi,
Have just returned from 3 1/2 glorious weeks in Taiwan. We flew with Cathay who I would recommend for those flying with kids (we have a 5 year old boy).
He spent almost the whole time using their console to play games/watch videos and didn't get bored at all. I'm not a big fan of kids playing games non-stop, but on a long flight I can tolerate it! There was enough variation to keep him entertained.
They also let an 18 month old kid (too big for a bassinet) sleep on the floor which meant he got a good nights sleep as well.
Service is good/friendly on board and there are free snacks throughout the flight.
At LHR they also let us check in at the Business desk because we have a kid (we were travelling economy).
Will travel with them again even if they were a little more expensive.