At that point in time the Qing Government was incredibly weak, couldn't France, the UK, Russia, Japan etc. just divide up China? Would the Chinese have been able to put up much of a fight?
What was stopping the West? Well, there's quite a few things, actually.
Firstly, the Taiping Heavenly Kingdom, which is easily forgotten in narratives of the Second Opium War (somehow). The legal ambiguity of the Taiping was always a little bit of an irritant, and there were also ambiguities in their treatment by Western powers. On the one hand, the terms of the 1858 Treaty of Tientsin included Nanjing as a treaty port, seemingly ignoring its being de facto owned by another state at the time. On the other, there was a prevailing opinion that, if left to their own devices, the Taiping would probably wear down the Qing in due time. The limited conflict of the Second Opium War could at least be considered to be maintaining a semblance of neutrality by simply raiding coastal areas removed from the areas of civil war, but a sustained war of conquest would be much harder to square internationally as something other than an outright intervention. Moreover, beginning a war of conquest against the Qing would very rapidly become a war against the Taiping as well, with the wealthy Lower Yangtze region being under their control. We must then also add the other rebel regions – the Nian between Beijing and the Yellow River and the Panthay Sultanate in Yunnan – to this equation, and ask whether it really made sense to try and establish a new set of foreign rulers over peoples already actively rebelling against their existing ones in favour of native rule.
The answer, as was recognised at the time, was no. Britain had just reasserted rule in India after the 1857 Rebellion (a.k.a. the Indian Mutiny) with the establishment of the Raj, and its confidence about its ability to retain effective imperial control over large Asian territories was severely shaken. Given how India was quite a divide et impera situation, how much worse would an attempt to establish order in China, traditionally seen as strongly centralised, have been? Moreover, the Indian population was then in the ballpark of 300 million, whereas the population of China was just over 400 million (450 million according to pre-Taping census data), an added hurdle to any attempt at direct rule.
Add to that the size of the military forces in China at the time. During the Opium Wars the maritime mobility of the Western powers was especially useful as it prevented the Qing from being able to concentrate their forces in time at the point of decision – at every battle bar two during the first war, the British, not the Qing forces, had superior numbers. However, during the Taiping War huge concentrations of men were in the field. Zeng Guofan's Hunan Army numbered in the realm of just over 100,000, and his intelligence reports suggested that there were 600,000 Taiping troops across six field armies (probably a pessimistic exaggeration), whilst the size of the other Qing Green Standard and militia forces is difficult to ascertain but probably would have been no smaller than those of the Taiping. In those kinds of conditions, small European forces numbering less than 20,000 would have faced an incredibly difficult situation indeed, especially as many of those Chinese forces, particularly the Taiping, had already begun integrating Western armaments by 1860.
The geography of China would also have complicated matters, especially if we're talking 'the whole of China' here. The major naval powers, Britain and France, would have had a hard time pressing inland with such a limited amount of land military presence (in neither Opium War were more than 20,000 soldiers deployed to China), especially to places like Sichuan (where the Nationalists held out against the Japanese in WW2) and the Tarim Basin, isolated from the rest of the Qing Empire by the Gobi Desert and the Himalayas; whilst Russia would have had difficulty on the coast against Anglo-French objections (although in the event the Russians did press territorial claims during this period, obtaining the north bank of the Amur as part of the 1860 Convention of Peking and nearly securing control of northern Xinjiang until the resolution of the Ili Crisis in 1881.)
That issue of mutual Great Power rivalry would have further compounded matters. Who should get what bits of China? The simple solution: Nobody. Let the civil wars resolve themselves and have any territorial concessions be small and mainly for administrative, diplomatic and commercial purposes within treaty ports or major cities. Hence the Beijing Legation Quarter or the Shanghai International Settlement.
However, all these hypotheticals detract from a key question: Why was there European interest in China? The answer was not gaining territory for the heck of it, it was for trade. Until the 1880s (when Japan industrialised) China was the dominant Asian economy. China imported silver, cotton textiles, weapons and luxury goods, as well as, most significantly of all, opium (although its priority in Western minds was limited – see note A), whilst Western powers purchased tea, silk, raw cotton, chinaware, lacquerware, furniture and paintings. Going to war in the tea-growing and silkworm-rearing regions on the Yangtze, Pearl and Yellow Rivers would have been an utterly counter-productive venture. It would have severely disrupted the production of what they were importing, and, crucially, destroyed the economy of the people they were exporting to. Far easier to do what they did in the end – secure ports of entry and maintain the availability of trade, and otherwise let the existing government remain in place. Hence the limited terms of the earlier Unequal Treaties up to 1860 (see note B) but also the intervention against the Taiping in the 1860s in order to accelerate the end of civil war on the Yangtze and enable the resumption of regular trade. The best proof of the inviability of maintaining territory in China would have to be the British occupation of the island of Zhoushan and its city of Dinghai from 1841 to 1846. The island was held hostage, in essence, until the payment of the indemnity specified in the Treaty of Nanking, but there was a move to try and keep it long term (it had been requested as part of the Macartney Embassy of 1793). However, insufficient customs revenues could be derived from what trade went through the city to cover the £70,000 annual defence budget, and so despite the tenacity of some individuals on the British side, the island was returned to Qing hands in the middle of 1846. Certainly the conditions of a major land occupation would have been quite different, but even so it is worth noting that administration is expensive, especially as the revenues being derived from the trade moving through the region would have been little different regardless of who controlled the regions of production.
So, in essence, they wouldn't because they couldn't, and any case thought they shouldn't.
Notes
Note A: The importance of opium
It is often easy to overestimate the extent to which opium motivated European imperialism in China. Certainly the opium trade bankrolled the British Empire for quite some time, but ultimately the main concern for politicians at home was the legitimate trade, especially in cotton textiles, as this trade, unlike the opium trade, employed significant numbers of British citizens at home, and was thus of more immediate concern. Letter-writing campaigns by industry during both the First Opium War and Taiping Civil War both emphasised the importance of keeping the British textile mills running and urging that the British government achieve a quick resolution to the situation. Moreover, there was always an understanding that the opium trade had to be relinquished at some point. Opium exports progressively decreased after the 1880s and were officially to be ended by 1917, though domestic production had exploded in the 1870s and would be a major source of revenue for many future regimes – the warlords, Manchukuo and yes, even the Communist Party.
Note B: The Unequal Treaties
What is often forgotten is that the first 'Unequal Treaty', the 1842 Treaty of Nanking, was ultimately almost identical to an 1835 treaty with the Khanate of Kokand. Both stipulated the payment of an indemnity, better communication between merchants and officials, extraterritoriality, renegotiated tariff rates, the end of the merchant monopoly at the existing point of contact (Canton and Altishahr, respectively) and most-favoured-nation status to the non-Qing signatory. Perhaps what made the later treaty palatable enough to be signed with little furore by the officials in Nanjing was that it had such precedents behind it, and certainly it derived to quite an extent from the more meticulously negotiated Convention of Chuenpi from 1841. Moreover, it must be added that the Qing Dynasty was never obliged to legalise the opium trade by foreign treaties, but rather appears to have done so voluntarily in 1858 as a method of raising revenue to fight the Taiping.
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