Good question. Despite their origins on the north-eastern frontier, the Qing Dynasty governed a fundamentally Han dominated empire. As such, the language of governance remained much the same it always had. Classical Chinese served for official written functions and anything concerning the imperial court. The Manchu language and script did not fare well during the Dynasty's rule. Successive generations of Manchu bannermen interacting with broader Chinese society (something frowned upon by the imperial government) led them to mostly adopt the northern Mandarin dialect, though this made them nearly as perspicuous from locals in most parts of the empire as the Manchu language had.

Manchu did receive imperial patronage, though not to the extent of requiring Han officials to learn it. Manchu bannermen were permitted to write their examinations for civil service in the Manchu script, and would receive favourable consideration for doing so. This policy would not however keep the script alive and the writing system had mostly disappeared by the end of the dynasty.

So why then, not simply require all civil servants to learn Manchu? Because fundamentally, no-one wanted that. The initial conquest had left the Qing in control of a enormous population that featured a large and somewhat intransigent literati class. Enough blood had been spilled over the issue of hairstyle and compromise had been needed over dress. Requiring the intellectual elites of the population to use Manchu instead of Classical Chinese would have been a pale too far.

But even after the conquest, Manchu remained a separate language for use by bannermen and the imperial court (usually only in private functions, mind) until displaced by court Mandarin. Successive Qing emperors held to the principle that the Manchu people, by virtue of their traditional way of living were morally superior to their subjects and thus held the right to rule. Thus the Manchu language was seen as a vital part of their exclusive traditions, though one increasingly discarded as time wore on.