Chinese cultural history would have to be completely different in order for the Qing to stay in power- no Xinhai revolution, no New Culture movement, no May Fourth movement, no Nationalists or Communists. The point of divergence would have to be really far back in order for them to even have a chance and modern China would be almost unrecognizable.
A 'traditional' Chinese state like the Qing can only stay in power if the royal house is in order and if China has the political and economic clout to maintain the Confucian diplomatic system. From the mid-19th century onwards Qing didn't have any of these requirements. The royal house was dominated by petty infighting and young, weak emperors, and efforts at reform were squashed from within the imperial household. Industrialization in the West had finally hit the point that Chinese economic power was collapsing. The Taiping Rebellion devastated China's most-productive region and killed millions while various regional revolts strained the army and the budget. And diplomatically, Vietnam was being conquered by the French, Japan had undergone a nationalist and anti-Confucian revolution, and China had been humiliated by the Opium Wars, the loss of Maritime Province and Sakhalin to Russia, and the expansion of spheres of influence in its cities. Toss on the Sino-Japanese War, which brought the annihilation of the modernized Chinese navy and the definitive loss of Korea, their last faithful Confucian ally, and the Qing is an utterly failed state.
This makes it tricky because prior to maybe 1840 it wasn't apparent that Qing would soon be in major decline, but once the First Opium War hit it was literally one problem after another until the dynasty finally collapsed. In terms of military or productive technology there was little that could have been copied until it was already too late (the iron-hulled warship Nemesis, which played such a major role in the war, had only been built one year earlier in 1839). It seems that Chinese defeat in these early wars mostly boiled down to human and organizational failures, not a large technical gap. The Qing army was poorly-led, poorly-trained, undisciplined, inexperienced and armed with poorly-maintained or hand-me-down equipment (in the Taiping rebellion and later wars they would be seriously outperformed by irregulars and citizen militias) while the British military was on the cutting edge, fresh from hard-fought wars in Europe and the colonies. At least since the time of the Qianlong Emperor, Chinese officials noted that their military was declining- if he or his successor had done an in-depth reform of the military to professionalize them and keep them well-supplied, then maybe that would have staved off some of the disasters of the coming century.