How would would defeat the Mongols if you were the Jin?

Oct 2019
Near the dogbowl
If the Jin took the Mongol threat seriously they could easily have stopped it prior to 1210; let's not forget that Chinggis Qan was still paying annual tributes at the border market until 1210, the Jin just had to kill him when he came to present tribute. The success of the Mongol attack in 1211 was unexpected as the Jin literally neglected their northern defences. For some unknown reason, the Jin did not try to maintain the old divide and rule strategy and allowed a large state to rise in Mongolia under Ong Qan after the Tatar rebellion of 1196. The Jin might have thought a relatively unified state might have been easier to manage and kept rebellion under check. When viewed retrospectively, the Jin should have just actively supported the different factions in Mongolia so Chinggis never attain hegemony. A serious frontier defense preparation could also have easily prevented an invasion and maybe even crushing the Mongol insurgency.
Pay the Mongols to attack the Sung? Is that possible? Otherwise your divide and conquer strategy appears best.
Sep 2016
Pay the Mongols to attack the Sung? Is that possible? Otherwise your divide and conquer strategy appears best.
Mongols would have to first either conquer Xi Xia or move through Jin territory. The former would be bad for Jin as it would empower Mongols greatly. The latter would also be bad, as that would mean allowing foreign armies to enter the core territory of Jin, leaving their cities and population open for pillaging or outright treason.
Feb 2018
Assuming this is post 1204 Mongol unification since otherwise the answer is rather obvious. Forget the Song, in 1209 Xi Xia asked for Jin help against the Mongols and the Jin Emperor told them to fend for themselves despite calling for war against the Mongols in 1208 [prior to his ascension] when Genghis Khan insulted him to a Jin envoy. A year later Xi Xia made peace with the Mongols and became their vassals, instead of a potential Jin ally. The Mongols were completely inept at handling fortifications mechanically or strategically at this point, and with Jin support it seems extremely likely that the Mongols fail to force Xi Xia to capitulate that year. That changes the entire strategic context of the war.

Ultimately the Jin lost 3 major battles in 1211, 1212, and 1213 that crippled their ability to match the Mongols in the field. While the details are not clear, they were all fought in mountain passes on the Jin choice of ground in 1211/1213 (1212 was an ambush possibly using the siege of Datong as bait). Evidently the Jin believed that this terrain, with their flanks anchored on the mountains, favored them, but the Mongols apparently took advantage of that assumption.

Gather all Generals, Captains and Lieutenants. Teach them how much the Mongolians love their feigned retreat tactic. Impress upon them the massive importance of not falling for that trick.
Other than the capture of Shenyang by Jebe, there are no major victories by the Mongols using the feigned retreat to my knowledge against the Jin in the first campaign of 1211-1215 (they used it against Xi Xia in 1209). In actuality the opposite happened in 1213-1214, when the Mongols dispersed their forces across the Central Plains and the Zhongdu defending army was paralyzed by a tiny army of 5,000 men in front of it (the threat of an ambush/feigned retreat, when in actuality the Mongol main forces were far away). In early 1212, when the Mongols retreated from the Zhongdu area, the major Jin army in the area did not aggressively pursue them either.