Diaoyucheng was part of the fortress system protecting Chongqing. Taking Chongqing was necessary for the Mongols to achieve their main goal of their attack on Sichuan: bypassing the formidable Yangtze River defences further downstream. Although in the end, the Yangtze River defences were breached by the Mongols at Xiangyang in 1273 (after a five year siege), years before any of the Chongqing fortresses surrendered, the Mongols felt that Sichuan offered a better chance. Thus, the early Mongol offensives against the Song focussed on Sichuan.
The Song were quite aware of this, and responded by a major fortress construction program, protecting Chongqing, Jiading and Kuimen. Möngke's 1258-1259 campaign's main target was Chongqing, but he had no success as far as taking Chongqing went (at least 6 fortresses were captured in the campaign, but none of the fortresses protecting Chongqing fell). Mongol losses appear to have been quite heavy even before the siege of Diaoyucheng (Rashiduddin wrote that Khubilai lost eight of ten tümens unsuccessfully attacking one city), from both battle and disease. Attacking Diaoyucheng only increased losses, with Möngke's death among them.
Why didn't Möngke retreat during the before or during the siege on Diaoyu Fortress?
Giving up on sieges when difficulties are encountered is a recipe for never taking strongly fortified places. Persistence is necessary, and the Mongols had learned this well. However, at some stage, it becomes wise or necessary to abandon a siege, or even the whole campaign. We don't know the campaign in great detail, but Mongol suffering and losses before the siege of Diaoyucheng appear to have been great enough so that abandoning the campaign before Diaoyucheng might have been the best rational choice. There are three likely factors that contributed to Möngke continuing the campaign, and continuing the siege:
Sichuan had been relatively quiet for many years before 1258. The Song had used this time to strengthen their defences - most of the fortress system was built during this pause in major Mongol attacks. Therefore, the Mongols had not yet grappled with the full strength of the new Song defences, and probably expected the defences to be weaker. The Mongols were probably guilty of "preparing for the last war rather than the next one".
Möngke's personal reputation as a commander was at stake. He chose to lead the campaign, and would be responsible for its success or failure. Wanting to be successful probably led him to accept higher losses than a subordinate general might have.
The "sunk cost fallacy": if he abandoned the campaign, or the siege, all of the losses incurred up to that point would have been for almost no gain. The capture of Chongqing would justify those losses.
Another factor might have been important, but it's unknown whether it might have entered Mongol strategic thinking. The final Mongol conquest of the Jin was very difficult because the area had been extensively fought over for many years. The devastation meant that the Mongols were normally only able to keep their armies in the field for a short time, not long enough to successfully besiege strong fortifications. Their alliance with the Song was essential: the Song provided logistic support that enabled the Mongols to fight for long enough (the Song also provided troops, which was useful, but the supplies they provided were more that just useful; they were necessary). Much of Sichuan had already been devastated before the 1258-1259 campaign, and despite the pause in major campaigns, still hadn't recovered. Repeated attempts to take the Chongqing fortresses would reduce Mongol ability to support their armies in the area, so capturing key fortresses on the first attempt would be useful.
As we know, Möngke's campaign failed. So did the next attempt, by Khubilai in 1263, and many attempts after that. The Chongqing fortresses started to fall in 1276, after the breakthrough at Xiangyang had already breached the main Song defences on the Yangtze. The remaining fortresses fell in the following years, with Diaoyucheng being the second-last to fall, in 1279 (it seems that Santaicheng outlasted the Song Dynasty, falling only in 1280).