In some ways, I think Partition was an attempt to retain India. I recall the film Gandhi where one of the British officers says something to the effect that there is a "Hindu India, a Muslim India, and an India of Princely States. And all these must be protected and cared for."
I really think that when Britain pushed Partition along with Mohammed Ali Jinnah (and boy, was he ever a piece of crap; a secular Muslim who drank and smoked himself to death and yet was one of the Indians most reponsible for Partition) they (the British) honestly thought that most Indians would opt for Home Rule, Swaraj, under the continued authority of the British Crown. They didn't realise how determined Indians really were!
I really think it shocked them when India truly told them to GTFO. I don't think they knew what to do. They knew then what you and I know now. India wasn't called "the diamond in the Crown of Empire" for no reason.
Britain knew that if they lost India, it was only a matter of time for the rest of the Empire. They got so much monetary wealth, and wealth in natural resources, out of British India that it greased the rest of the Empire! I mean, Crap, just in taxation alone they got more out of India than anywhere else in the Empire.
The Brits were damned clever.They could NEVER have ruled India by conquering it in the traditional way.So Divide and Conquer, as you said. After the British East India Company was removed and the Raj was instituted in 1858, they were smart enough NOT to interfere in local governments. They did what previous Empires, such as the Moguls, had done. They set themselves on top and simply served as the highest level to which all taxation went, as well as serving as the final court of appeal judicially and so-forth.
But they always knew that if the System ever broke down, the whole Empire was screwed. Because they did things one better than the other Empires simply with that British efficiency for which your country has always been so well known. They outpaced the Moguls and the others. And it worked. But it was what greased the rest of the Empire. If it failed, they would no longer be able to afford to keep the rest of the Empire without going bankrupt.
And that is what they were headed for after India left. Which is why by 1997 there was practically nothing left when Hong Kong left aside from some outlying areas that would never survive on their own.
Granted, you know this as well as I do, being an Englishman. I know it because my BA in History and my MA in History are BOTH in British History. And I studied the Raj in considerable detail. I also have a BA in Philosophy but is beside the point.