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.

歷史上有很多精彩而富有創(chuàng)意的回答,因此我想提供一個更傳統(tǒng)的、基于真實歷史的分析。

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?

宋朝被蒙古滅亡更像是一系列偶然事件和愚蠢政治決策的結(jié)果,而不是因為蒙古軍隊真的不可戰(zhàn)勝。事實上,宋蒙戰(zhàn)爭(1235-1279年)持續(xù)了34年,大部分時間都是拉鋸戰(zhàn)和持久戰(zhàn)。但究竟是什么突然打破了戰(zhàn)爭的平衡?

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,忽必烈活得特別久。
通常,一位可汗去世后,草原部族之間會發(fā)生內(nèi)戰(zhàn),從而削弱蒙古的力量,這對宋朝是有利的。如果忽必烈沒有活得這么久,宋朝可能像抵御契丹和女真一樣繼續(xù)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,襄陽之戰(zhàn)中宋將呂文德早逝。
呂文德是襄陽之戰(zhàn)的關(guān)鍵人物,他原本成功擊退了蒙古的多次圍攻。但他在圍城戰(zhàn)開始的第一年就因病去世,由能力遠(yuǎn)不如他的弟弟呂文煥接替。后來呂文煥在城中仍有足夠糧食支撐十年、守軍幾乎完整的情況下選擇投降。如果呂文德沒有早死,蒙古幾乎不可能攻破襄陽。

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,宋朝當(dāng)時的宰相賈似道不愿全力增援襄陽。 
他擔(dān)心一旦離開京城,政治對手會趁機(jī)取代他。但如果他不親自領(lǐng)軍增援,誰也無法調(diào)動大軍。他整整浪費(fèi)了六年時間,直到襄陽最終陷落。

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,水軍名將劉整的叛變。  
劉整后來幫助忽必烈建立水軍,并在戰(zhàn)爭中首次擊敗了宋軍水師。他的叛變正是由于賈似道的愚蠢政治手段。如果沒有劉整,忽必烈在多次失敗后早已失去征服宋朝的興趣,尤其在他兄長蒙哥汗去世后更是如此。而且,如果宋軍水師能繼續(xù)保持對蒙古水師的優(yōu)勢,襄陽的水路就不會被切斷,襄陽也不會陷落。

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

所以,如果我是當(dāng)時的宋帝(即宋度宗),我會:(按優(yōu)先級)

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. 派遣刺客刺殺忽必烈,此人是蒙古內(nèi)部穩(wěn)定的關(guān)鍵人物;
2. 罷免賈似道,防止劉整叛變,或提前處決劉整;
3. 在襄陽被圍之初就親自領(lǐng)軍全力增援;
4. 用更有能力的將領(lǐng)替換呂文煥。