There are two ways in which Kadima and Yisrael Beitenu will be members of the same coalition. Either under Livni or under Bibi. As I’ve already said, the only way Livni can form a coalition is if Lieberman sells out. Yes, Bibi could join a Livni-led coalition, but if Lieberman holds his own, I don’t see that happening – Netanyahu has a stronger hand right now.
The other, only slightly more likely, possibility is that Netanyahu actually manages to convince Livni to join a government under his leadership. Any future government in this Knesset will almost surely include Avigdor Lieberman, as a senior member of that coalition. And since Netanyahu has already claimed he wants to form a the largest possible coalition, we might see all three parties in the same government (total number of seats of those three: 70). I do not think this will happen, for reasons I will explain, but in case it does, it might be the best option.
Traditionally, in Israel, the government is very ineffective. I don’t see this changing any time soon. However, Israeli government have also usually been very damaging to Israel. Thus, if Kadima joins a Likud coalition, there will be so much infighting that the government will probably be, just as ineffective, but will also be relatively powerless – too hamstrung to do any real damage. The status quo will remain the staus quo, and I can think of much worse things than that.
Nevertheless, this scenario is not very likely, Livni does not seem as durable of a figure on the Israeli political landscape as, say, Bibi, Peres, Barak, etc. If she does capitulate, and join a Likud-led government, her days as chairwoman of the internecine Kadima will be numbered. She stands a far better chance of “long-term” survival is she pretends to stand up to Netanyahu.