There's no evidence for the statement that 'nations were all fractured and weak'.

The best way is to trace the phases of Mongol conquest from Genesis Khan to the post Kublai era.

Starting with Genghis Khan. His 2 biggest foes were the Jin empire in Northern China and the Khwaremzian empire of Mohammed II. Both were united, well organized, richer and well organized 'nations'. Indeed it's a reasonable thesis that the Khwaremzian state was the richest and most powerful state in the Europe to Asia region (I dont know enough about Africa and the Americas). It was significantly materially stronger than the caliphate based in Baghdad.

The post Genghis to Kublai phase was dominated by the conquest of Russia and the capture of the middle east including Baghdad. Yes, the Russians were organized as a group of principalities and this did lead to their defeat. But they weren't weak. They were undone by Mongol preparation and speed and mobility. You could argue that they never learned the lessons of the Battle of the River Kalka and the Subudai/Jebel raid but that's a fundamental issue with political structure - not weakness. Remember that the Russian principalities did offer a combined response at the River Kalka. So they could come together when the situation warranted. Similar thing happened with the Ghurid invasions of India. Hungary was definitely not a weak state. It did have a struggle for balance between the King and the nobility (Boyars) but the Hungarians did reply in force. They were simply outclassed in terms of preparation, mobility and the use of Intelligence. Hulegu's capture of the middle East probably can be called a sweep of weak opposition but the wealth of Baghdad shouldn't be underlooked just because he made it look easy.

Kublai's conquest of the Southern Song was extremely hard fought at the borders of the empire. The Song empire was strong, united and prepared. They just had crappy leadership compared to the Mongols. Once the Mongols breached ther borders - brilliant execution by the Mongols under Mukhali and weak Song leadership finished the Song.

Post Kublai - Yes the opposition was weaker but that's because there were few states left to take on. Interestingly in this phase the Mongols failed to defeat the Khilji dynasty in India despite repeated forays. They encountered strong, prepared opponents and some brilliant military leadership that was able to neutralize traditional Mongol strengths.

I don't think the thesis that nations were unusually fractured holds.

Rather than asking which of those regions (Eastern Europe, the Middle East, China, and Central Asia) were fractured, it's easier to ask which were united. The answer: none of them.

The Khwarezmian Empire was nominally united, but much of it was recently-conquered, and minimally loyal to Muhammad Shah. Muhammad Shah's conquest to build his empire were only completed two years before the Mongol invasion, and the empire was still fragile. For example, Samarkand had revolted 7 years before the Mongol invasion, and been sacked by Muhammad in response. This lack of real unity was probably a major factor in Muhammad's defensive plan against the Mongols of using his army to garrison his cities, rather than seeking to defeat the Mongols in open battle. Even with this plan, many cities surrendered quickly to the Mongols, some without fighting, and others after resisting for a few days. (Some cities showed no more loyalty to the Mongols, either, revolting as soon as the Mongol armies had moved away, much to their own misfortune, as this elicited a brutal response from the Mongols.) This passive defence gave the Mongols the strategic and operational initiative, helping the Mongols in their conquest.

China was a much more stubborn opponent, but was far from unified. The Mongols entered what had been a three-way war for the control of China, mostly between the Jin, holding Manchuria and most of northern China, and the Song, holding Sichuan and southern China, with Xi Xia (the Tangut Empire) as a minor player in the north. It took the Mongols 23 years to conquer the Jin, who continued to resist for almost 20 years after they had lost Manchuria and Beijing to the Mongols, and the Mongols needed assistance from the Song (mostly logistics, but also troops) for their final victory over the Jin. After that, it took the Mongols almost 40 years of warfare, culminating in the five year Siege of Xiangyang (1267-1273) to break through the Song defences, and then a further 6 years to defeat the Song. Had China been united, it is quite imaginable that the Mongols would have made little progress. Even the Jin not fighting the Song, and not struggling with internal revolts (in both China and Manchuria), would have made the Mongol task much more difficult.

Much of the Middle East was a patchwork of small states (including the Crusader states). The largest single power was Egypt, which stopped Mongol expansion in the Middle East through a series of victories on the battlefield (it wasn't just a singular victory at Ain Jalut). Eastern Europe and Russia were likewise divided.

Not coincidentally, Mongol expansion stopped as the Mongol empire divided into separate khanates, at first nominally part of a united empire but practically independent, and later becoming fully independent.