I think Leia would have sought out Obi-Wan. Obi-Wan would have then taken Luke. Since Vader is dead, there would be no reason to hide that they're brother and sister.
Obi-Wan would have trained them both, together.
The 3 of them would travel to Dagobah, and train more, under Yoda.
Together, all 4 would eventually try to build the rebellion through other means. Ultimately, Starkiller would be sent to destroy them, but his guilt for the actions on the death would cause him to turn to the rebellion's cause (keep in mind, he didn't join the emperor by choice in that ending).
The 5 of them would go to face the Emperor, under the facade that they were starkiller's prisoners. A battle would break out.
Ultimately, Luke or Leia would be the ones to kill the Emperor. Then Yoda would realize that the prophecy was true... Anakin did bring balance to the force, but not by killing the emperor himself. He brought balance by marrying Padme, having Luke and Leia, who ultimately killed the emperor, turned Starkiller to the goodside, and eliminated the Sith.
Luke, Leia, Starkiller, Obi-wan, and Yoda would then make up the new Jedi Council, and the rebirth of the Republic would begin.