At the height of the British Empire, why did it not conquer Europe?

Greetings! What a most interesting question, and one which does not actually have a proper "answer" in the sense of a "this old chap, is why we didn't decide to conquer those bloody continentals". There were a variety of interesting reasons why the British did not conquer Europe, though they certainly formed ideas about how the British Empire ought to deal with the European nations throughout the nineteenth century. Speaking of which, it should be noted here that "the height of the British Empire" as OP has put it in their question, is definitely a time period up for interpretation, but the "Pax Britannica" of 1815 - 1914 seems a good time period to explore. Let's begin.

Europe and The Empire

"“[France’s greatest goal] was the humbling of England, the traditional Rival of France, and the main obstacle to French supremacy in Europe and all over the world.”

- Lord Palmerston during his time as Home Secretary under the prime ministership of the Earl of Aberdeen

When the "Great War" of the 1800s had finished in 1815 (almost a century before another, far larger one would break out), Britain had emerged as the supreme victor over the forces of Napoleon. She had resisted the French threat of a cross-channel invasion, used the resources of her growing empire to survive the (poorly enforced) Continental System, and her armed forces had distinguished themselves over two decades of tumult in Europe. Perhaps most critically of all however, Britain had won from the Napoleonic Wars a guarantee against European threats. In the "Concert of Europe" which followed the Congress of Vienna, the great powers of Europe pledged that they would maintain the status quo in territorial matters, and that any hostile attempts to gain more land by one power would be checked by the threat of war against all the others. John Darwin on what this meant for the British:

"For almost a century, the concert principle held - to the huge relief of the British. The supreme threat to their empire - a Europe united against them - receded into the shadows."

London's Foreign Secretaries and Prime Ministers continued to follow a general maxim laid down by the great diplomats of Lord Castlereagh and George Canning in the early 1800s: Britain would keep an eye on Europe, but seek to avoid all involvement on the continent which did not directly threaten her own national security, or that of her empire. It was such thinking which influenced British involvement in the Crimean War, allying with France (and a reluctant Austria-Hungary) to see off the Russian threat to the Mediterranean and the overland route to India.

Yet why not invade Europe? The question commands some serious thinking. If the Royal Navy controlled the sea lanes and earned the ire of every leader who spied a British fleet off their coasts, then why did the British government not send expeditions to Europe in much the same way they had dispatched troops to quell the Sepoy Mutiny or to blast open Qing China during the Opium Wars? Surely, if the British sought to diminish the gravest threat to their empire, then was direct conquest not the most effective way of neutralising the other Europeans?

For two key reasons, this was not possible. The first is more practical and economical: funding a European war. In the Napoleonic Wars, Britain had served as banker and paymaster to its allies in the Grand Coalitions, ensuring that they maintained the economic strength to continue fighting Napoleon's French Empire. If Britain were to invade Europe, then it would find itself against the French in addition to multiple other powers (recall the Concert system and its conditions for hostile expansionism discussed earlier). The cost of financing a war against multiple highly modernised, effective, and well-led armies in Europe would all but break the City of London's financial capabilities. It was also particularly problematic that, of the great powers, Britain was actually the least prepared for war at a moment's notice. After the "imperial peace" of 1815 and the Crimean War of 1856, the British were the only great power in Europe without a general conscription law. It was certainly something the government liked to show off as proof of British democracy and individual freedoms, but it also meant that mobilising an army in wartime would require disproportionately more time and effort than Prussia (Germany), Russia, France, and Austria-Hungary (where reserves could be called up in a matter of weeks, and in some cases even days). A war in Europe would be particularly troublesome to win, if not downright impossible.

Let us then turn to the second consideration, which is in a way tied to the first. If Britain were indeed capable of conquering Europe, then controlling it would be more difficult. Across the Empire, the British had been able to cement their rule through alliances with local leaders, or through "cession treaties" which granted formal control from a prior holder of power. In Europe, the highly centralised and constitutional (not to mention rising national) polities would likely be extremely uncooperative, and at the very least highly resent British rule. The thought of aristocrats from Britain taking over the landed estates of the Junkers in Prussia seems most amusing, but in reality such a situation (amongst many others in the process of implementing 'imperial rule' in Europe) might have led to open revolt and rebellion, something the British absolutely abhorred within their administration of the empire. Active diplomacy, not armed campaigns, was Britain's byword in the "Pax Britannica" period of the 1800s. They maintained a reluctant (and at times rather condescending) hand in European great power politics, but preferred to keep the "balance of power" strictly within the boundaries which best favoured their continued imperial prosperity and "empire economics".

When the Third Derby-Disraeli Ministry formed in 1866, this idea was entrenched even deeper in the minds of the British cabinet and government. Their empire was the first and primary concern, and (as some more imperialist elements argued), it should indeed have been the only thing British foreign policy was concerned with, to the exclusion of all others. Under this Conservative government, The Right Honourable Earl of Derby, or Lord Stanley served as Foreign Secretary. The Austrian Ambassador to Britain, Count Rudoplh von Apponyi, gave a rather insightful overview into the new Foreign Office and its vision of the British foreign policy which helps explain why they did not conquer Europe.

“[Lord Stanley] belongs to that school of statesmen who make a dogma of the most complete non-intervention and the most absolute abstention of Great Britain from the affairs and quarrels of Europe. According to Lord Stanley, the only great interests of this country consists in the pacific [peaceful, not the ocean] development of its prosperity and its colonial and commercial power, and, as England could not attain this goal and at the same time interfere actively and influentially in the affairs of Europe, she ought not to hesitate between the two courses.”

William Gladstone, another architect of British foreign policy in the latter half of the 1800s, also eschewed a similar set of ideals, remarking on the British commitment to Europe:

“England should keep entire in her own hands the means of estimating her own obligations upon the various states of fact as they arise … that it is dangerous for her to assume an advanced, and therefore an isolated position in regard to European controversies; that come what may, it is better for her to promise too little than too much.”

Thus to conclude, the British Empire did not conquer Europe for both the practical reason that such an invasion would have been disastrous for its economy and military (to the point where such a war would produce little gain for unacceptably high costs), and also the geopolitical reason that the 19th century "Concert system" had neutralised the threat of a united Europe, and also provided the pretext for Britain to turn towards the security of its Empire from individual states rather than an entire continent of rivals.

Hope this response helps, and feel free to ask any follow-ups as you see fit!