Ailecek means "with all family/with whole family".
It's my opinion, I think that 'Ailece' is sounds more correct, but I checked TDK (Turkish Language Institution) it says both are correct. And both means same.
Ailece/k;
Aile(Family)
-ce/k(with, whole)
Here is some examples to understand the postfix(-ce/-ca/-çe/-ça);
Okulca geziye gittik. (We went to trip with (whole) school)
Sınıfça yemek yiyoruz. (We're eating with (whole) class)
Ailece/k pikniğe gidiyoruz. (We're going to picnic with (whole) family)
Hope that was helpful for you)