Did Great Britain benefit economically from the loss of America?

Two very good answers while I was thinking about this and trying to find sources. I cannot match the scholarship of either of the other answers, but I want to provide some contextual backdrop. These are some lessons I've learned over my years of studying the American Revolution; things that surprised me because I hadn't put them together.

Great Britain had just finished the Seven Years War (known in the colonies as the French and Indian War). This was arguably the first world war - pretty much every nation that had the ability to project power beyond their borders was involved. Although the war was over, each of the nations retained the desire to create, maintain and exploit colonies. (There is an easy book here on the history of empire). 1. Britain acquired an Empire. Defeat of the French on North America (Quebec) can be considered the beginning of the British Empire. There was a huge difference in the obligation to protect colonies against indigenous forces vs the requirement to protect colonies against other superpowers. The French/Indian alliance also made it clear that although the American Indians didn't have artillery, they could still participate in logistics and in what would later become known as asymmetrical warfare. 2. Britain acquired the deficit spending that comes with Empire. The seven years war cost Britain so much money that the resulting economic crisis lasted until the Napoleonic wars.
3. Britain had to demobilized a huge military force. Although they didn't have modern economic theory, they knew that demobilization can lead to social instability. Furthermore, every officer had to be placed on half pay or retired. (Everyone with an interest in history should read the Aubrey/Maturin novels that detail Jack Aubrey's pathetic existence on half pay; it sounds like a good solution, but it wasn't.) One solution was to move the most valuable units to North America, use them to defend the colonies against the Native Americans and as a guard against the colonial ambitions of France (et. al.). The nice part being that the colonies could be convinced (forced by fiat) to pay for the support of the troops. 4. Remember that the Glorious Revolution wasn't that long ago. England hadn't yet finished designing the parliamentary government. Issues like the King's right to a standing army, and "taxes are the free gift of the people to the government" were far from settled. The King and Parliament were not arguing from a particularly long precedent, and precedent is much more important in the English "pure common law" system than it is in the American constitutional system that we have today.

One last point - OP asks if England benefitted economically; I think that is an unanswerable question. At the time England's economic policy was mercantilist. Mercantilists understand economic benefits in a way that doesn't map very well to our modern notion (informed by Smith, Keynes et. al.) I think it is safe to say that shifting the costs of troop maintenance to the colonies seemed to Parliament to be a solution to a problem. I think it overstates things to say that this was a policy.

While the UK 'lost' the 13 colonies, it still 'possessed' Canada. Therefore it still had trade opportunities and defense commitments in North America, and similar arrangements in Australia and New Zealand. The character of these in particular is that they were sparsely populated by aboriginals, they were at a significant distance from the 'mother country', and they were colonized largely by 'their own people'. India doesn't fit this description - India already had both advanced social orders and had been 'civilized' for thousands of years.

While the colonies were 'revolting', the British had other things on their mind, in particular the Industrial Revolution. Many of their more immediate concerns was the construction of steamships and railroads, dealing with the social costs of rapid industrialization, and restructuring the political order to deal with the emerging merchant and professional classes. In this respect Britain was trying to 'straddle the fence': they had both an external and internal focus. The colonial adventures were an attempt to secure raw materials, which often didn't work out as planned.

Britain granted Canada 'independence' in 1867 - without much more than a ceremony and signing some agreements. In short, it didn't take even another 100 years to realize that attempts at ruling North American territories wasn't worth it. Much of this would have been due to the massive investments needed to, for example, built railroads across 3000 mile expanses of wetlands, plains, and mountains. While resources were there, manpower wasn't. Populating North American involved massive waves of immigration from nearly every other country on Earth, a process that continues to this day.