There's a Chinese on the way out of town, past the new port and maybe 1 mile up on the left. It's on the Disco Strip. It sits back a bit so is easy to miss. It's actually Chinese & Mexican, called Chopsticks & Chilli, but don't let that put you off. It's actually quite nice and the best Chinese I've eaten on the island.
I've had better in the UK and I have a particular passion for Soft Shell crabs in Peppercorn Salt but Chopsticks is very good for Corfu.
I'm not sure it's open at Xmas but others here will confirm. I hope it is as I plan a visit when we are down this year.
Corfu town has a plethora of good eateries and being Corfu Town the majority will be open. It's only the tourist areas that close completely.
There is an excellent Italian on Guilford Street. You walk up Guilford past the large church, cross a fairly main street of cobbles and about another 50 metres on your left is the restaurant right opposite a little church. I can;t remember the name but the Sword Fish carpaccio is to die for.
I've assumed you are going at Xmas but your post says winter but no doubt the places I mentioned will be open throughout the non tourist season.