I have not flown with sick kids myself (you're braver than I am!) but I do have two pieces of advice that might help that I got from other, more worldly travelers.
- Give the child a bottle/straw cup (depending on age & ability) during takeoff (and, to a lesser degree, landing). The sucking and swallowing will help their ears more easily adjust to the rapid altitude shift.
- Prior to takeoff, spray both nostrils with a nasal spray (Little Remedies makes one called Little Noses, and there's also a number of companies that make a saline spray, if you don't want to use chemicals). The altitude change drives the spray farther up into the sinuses than the spray can do on its own, and helps throughout the flight.
That second one was a tip from my jet-setting boss, back in my pre-kid days. I know it's worked well for several adults, but I haven't tried it with kids. The physics involved should be the same, though.