The Mongols, infuriated that the Song were trying to leech of their success, declared war on Song in 1235 AD. By 1248 AD, when the Mongols and the Song negotiated a ceasefire (ending the First Phase of the war), the only major gains the Mongols had were Chengdu.
At the same time, the Mongols had sent a sizable force between 20,000–35,000 men to conquer the Rus’, which was achieved through direct conquest of certain states and vassalization of the rest. While the Mongols viewed the Chinese as formidable opponents, they didn’t take them as a serious threat to their rule in Northern China and sent a portion of their troops to destroy Russia. While the Russians were also strong opponents, they didn’t have nearly as much experience fighting nomadic warriors as the Chinese had, for the Chinese had spent the last millennium trying to resist the nomadic horsemen who constantly raided their northern provinces. They also lacked the numbers that Song China could produce.
By 1241 AD, as the majority of the Mongol army and its Chinese conscripts waged war against the Song, the Hungarians fielded their entire army at Mohi to confront the Mongol host that was rampaging through Europe. At Mohi, the Mongols cut down tens of thousands by hailing arrow storms down on Hungarian infantry and knights.
One of the main reasons why the Mongols invaded Europe while still fighting China was because of the type of warfare the Chinese waged. Chinese defenders tend to stay behind the walls of their castles and avoided directly engaging the Mongols unless they were on the offense and tried to avoid exhausting themselves. The Europeans, who never had dealt with nomadic warriors since the Magyar invasion of the 900s, had no idea what to do and got slaughtered. For the Mongols, the initial invasion of Europe was a cakewalk due to European inexperience to deal with the Mongols.
Hope that helped!