The best way to have the Jin to have all of China
1. March south after the Jingkang Incident. This would ensure that the establishment of a Southern Song would be smothered in infancy.
2. Victory in the Jin-Song Wars before the appearance of the Mongols. This would take the Manchu defection system, but doable if you keep the Mongols from getting ambitious - not many nations in this world can successfully win a two-front war.
3. A successful alliance with Western Xia.
Emperor Renzong of the Western Xia made overtures to the Jin.
[In 1170, Renzong discovered a plot to kill him. He executed the generals who were behind the plot. As a result, Renzong distrusted his army generals and the army began to fall into incompetence. During his later years, Western Xia began to fight wars against various enemies leading to the eventual defeat of the Western Xia.] Ha there been no assassination attempt, the Xia would be better prepared to fight the Mongols.
Genghis Khan and other Mongols often led attacks on he Western Xia under Renzong's heirs. An alliance between the Xia and the Jin could in theory stop / delay the expansion of Mongolian influence. Or... a lucky archer could kill Genghis in one of his attacks on the Western Xia - with Genghis dead, the Mongols would be too busy fighting among themselves to be able to be able to conquer anything.
With the Mongols nerfed with the Western Xia alliance, the Jin could focus south.