It seems to me that India is one of the most culturally heterogeneous major nations with populations having vastly different religions, cultures, and languages. The only reason there was ever a unification was by force under the British crown. Many regions I know of with a history of incorporation by force seem to have thriving separatist movements (e.g. Quebec, Chechnya, Scotland, Tibet). Look at Spain for example. Even though Spanish is the dominant language, it seems each different culture has a thriving separatism movement. Why is India held together so well, especially considering the government doesn't seem to have strong control over many parts of the country?
First off, despite the general assumption, Britain didn't unite India.
If you look at this political map of India at 1947 (the blue areas are British-administered)
https://www.reddit.com/r/MapPorn/comments/3w8a0y/india_before_1947_1600_x_1450/
you can see that Britain's Empire was not much different from the previous empires of India i.e a strong central power ruling over weak vassals. At that time, India still had around 583 princely states who were nominally vassals of the British Empire.
The credit for the political integration of India, goes to India's first Home Minister Sardar Vallabhbhai Patel, who was ably assisted in his task by his secretary V.P. Menon. This wiki article is a good start if you are interested, although V.P. Menon's "The Story of the Integration of the Indian States" is better.
As for the separatist movements, there have been and are several separatist movements.
https://en.wikipedia.org/wiki/Separatist_movements_of_India
The Indian govt, after battling separatist and insurgent movements since 1955, seems to have settled on a policy regarding such movements, after making many mistakes.
A) Ban and use security forces against those organisations which explicitly call for the dismemberment of India by using violence, but not against those who want to do so non-violently or democratically.
For example, terrorist groups like Lashkar-e-Taiba or Jaish-e-Mohammed which call for the creation of an independent Kashmir by use of violence are banned, but the All Party Hurriyat Conference (APHC) which has the same goals but intends to achieve it via peaceful means is not.
Similarly, the Communist Party of India (Maoist) or CPI(Maoist) which calls for a revolution using arms to overthrow the Indian govt. is banned, but the CPI (Marxist) which works to achieve its aims by democratic means is not. In fact, it was a coalition partner in the preceding UPA govt.
B) Conduct free and fair elections under international supervision in the affected region, and accept the verdict of the people even if its against India's national interests.
For example, the present J&K Chief Minister is considered a "soft separatist", yet she is power due to a coalition with the BJP which is nationalist.
C) Increase efforts towards the economic development of the affected region via special economic packages or policies.
Well for a start, Pakistan and Bangladesh both used to be part of India, and both separated as the result of separatist movements, so I'd definitely challenge the premise of your question. Furthermore India existed as a unified entity long before the British Empire conquered it.
What makes you think there aren't? Kashmir and the tribal regions in the North-East are constantly plagued by separatist insurgencies.