Livni wants to be the PM not to be second in command helping somebody else win an election which is why she has so far refused all offers to join other parties and decided to go it alone.
If Livni was going to join an existing party it would be Labor because she knows that if she is a member of Labor in future party primaries she may be able to topple Shelly as leader so she has a chance at one day being PM.
Lapid is the last person whose party she wants to join because that party exists so he can have a political career.
Livni knows that if she were to join Lapid she would never get a shot at being PM one day because her job would be to help Lapid become PM.
If Livni forms her own party alot of the seats will come from Labor and Lapid so now they all want to be her to be on their team as number 2.
The only one who doesn't want her is Bibi because she is an asset if she if she splits the Labor / Lapid vote.
I imagine Livni wants to form a party this time, show everybody she would make a good PM, hopefully join a coalition.
Develop some credibility and increase her party membership then next time hope she can get enough support to win an election not help somebody else win.