I think Kate is the daughter of Hua Sun with tony not with jack.
Once, Hua Sun asked Tony to lover Kate as his real daughter and take good care of her if she was not around.
Second, Peter said at the birthday party of Aunti Mii that children were good at drinking because they got the gene from their father. And sure that is a reference to the hidden secret of Kate's drinking skill which is said to be as good as Tony's.
Third, real truth is with that Yuk Lai Mek. We actually don't know who gave that to Hua Sun, and before she gave that Yuk to Kate, she said something like "May the love of mother and father protect you", but she didn't look at Jack though he was the one who wore that YUk on kate's neck. That didn't show the love of her toward her husband nor the father of her daughter. More suspicious is when she was with Jack that night, she was drinking for sure. But she did have her sense and knew what she was doing with Jack. Just before she became Jack's girl, she took off that YUk Lai Mek and put it under a pillow like she was trying to avoid that YUk, meaning to avoid the real owner of the Yuk.
In over all, i think that that Yuk is given to Hua Sun by Tony and Tony is the real father of Kat.
What do you guy think?