Zhang Zhiheng
There are so many good and innovative answers, so I think i would provide a more traditional answer based on what really happened in the history.

历史上有很多精彩而富有创意的回答,因此我想提供一个更传统的、基于真实历史的分析。

The fall of Song Dynasty to the Mongols was more like an accident and a result of a series of coincidences and stupid political decision-making. Mongol army was not invincible and the war between the Song and the Mongols(1235-1279) had lasted for 34 years which means most of the time it was lean and tiresome. But what suddenly broke the balance of the war?

宋朝被蒙古灭亡更像是一系列偶然事件和愚蠢政治决策的结果,而不是因为蒙古军队真的不可战胜。事实上,宋蒙战争(1235-1279年)持续了34年,大部分时间都是拉锯战和持久战。但究竟是什么突然打破了战争的平衡?

Firstly, Khublai Khan lived accidently long, much longer than his other ancestors and offsprings, normally there would be a civil war which divided and weakened the nomad tribes after one Khan’s death which would benfit Song Empire greatly. If Khublai Khan was not significantly long lived, the Song empire would survive as same as the way they had survived through Khitan and Jurchens aggressions.

首先,忽必烈活得特别久。
通常,一位可汗去世后,草原部族之间会发生内战,从而削弱蒙古的力量,这对宋朝是有利的。如果忽必烈没有活得这么久,宋朝可能像抵御契丹和女真一样继续存在。

The Song general of the battle of xiangyang, which decided the fate of Song, Lv Wende died because of illness in the 1st year of the 6 years’ siege and was replaced by his far less capable younger brother Lv Wenhuan, who decided to surrend to the Mongols later, when there was still enough food for another 10 years’ siege in the city and the defence forces were almost complete. Lv Wende had fought the Mongol for years and was proved to be very capable of defending xiangyang by defeating the former Mongol besiege, if he didn’t accidentally die in time, the Mongol had zero chance to breach xiangyang.

其次,襄阳之战中宋将吕文德早逝。
吕文德是襄阳之战的关键人物,他原本成功击退了蒙古的多次围攻。但他在围城战开始的第一年就因病去世,由能力远不如他的弟弟吕文焕接替。后来吕文焕在城中仍有足够粮食支撑十年、守军几乎完整的情况下选择投降。如果吕文德没有早死,蒙古几乎不可能攻破襄阳。

The PM of Song at that time, Jia Sidao was reluctant to fully reinforce xiangyang because he was afraid if he left the capital his political enemies would use the time to replace him, but if he didn’t lead the reinforcement by himself, no one could. He had wasted 6 years waiting there until xiangyang finally surrended.

第三,宋朝当时的宰相贾似道不愿全力增援襄阳。 
他担心一旦离开京城,政治对手会趁机取代他。但如果他不亲自领军增援,谁也无法调动大军。他整整浪费了六年时间,直到襄阳最终陷落。

The rebellion of the famous Navy general Liu Zheng, who later helped Khublai Khan build his navy and the defeated Song Navy for the first time in the tens of years of war, was a result of Jia Sidao’s stupid polical schemes. Without Liu Zheng, Khublai Khan had no more interests to conquer Song anymore after his repeated failure and his own brother Menge Khan’s death. And if Song Navy could maintain its superiority as it always had to Mongol navy, there would be no way to cut off xiangyang’s water way and xiangyang would never fall.

第四,水军名将刘整的叛变。  
刘整后来帮助忽必烈建立水军,并在战争中首次击败了宋军水师。他的叛变正是由于贾似道的愚蠢政治手段。如果没有刘整,忽必烈在多次失败后早已失去征服宋朝的兴趣,尤其在他兄长蒙哥汗去世后更是如此。而且,如果宋军水师能继续保持对蒙古水师的优势,襄阳的水路就不会被切断,襄阳也不会陷落。

So if I was the Song Emperor then, 宋度宗 actually. I would :(By priority)

所以,如果我是当时的宋帝(即宋度宗),我会:(按优先级)

Send assasin to murder Khublai Khan, this guy was an outlier which kept Mongol from its nomad circle of civil war
Remove Jia Sidao from PM position to avoid the rebellion of Liu Zheng, or just kill Liu Zheng before he managed to rebel.
Led the reinforcement by myself at full strength at the beginning of the besiege of xiangyang
Replace Lv Wenhuan with someone more capable

1. 派遣刺客刺杀忽必烈,此人是蒙古内部稳定的关键人物;
2. 罢免贾似道,防止刘整叛变,或提前处决刘整;
3. 在襄阳被围之初就亲自领军全力增援;
4. 用更有能力的将领替换吕文焕。