How significant was the role of 'violence' in the Indian independence movement?

It's not clear what you're asking. Are you asking how much violence occurred in the independence movement? Or are you asking how important that violence was towards driving the independence movement, or how importantly it factored into the British decision to leave India?

In general, the Indian independence movement was remarkably non-violent. One reason was the powerful influence of Gandhi, who championed non-violence (ahimsa) and was bitterly opposed to violent confrontations with the British. He led peaceful marches which included tens of thousands of people who were specifically instructed to not offer any resistance to the police and to allow themselves to be arrested when challenged. The goal, of course, was to overwhelm the British, on the assumption that the country did not have the means or resources to jail millions of people.

More broadly, it was part of his ideology of non-cooperation. The reasoning was based on the fact that at the height of the British Empire, there were only about 20,000 Britishers in India, controlling a population of about 400 million people across the Indian subcontinent. For every British man, woman or child in India, there were 20,000 Indians. As such, control over the Indian population was only possible through the active cooperation of Indians, including millions of Indians in the police and armed forces, as well as the administrative bureaucracy.

Gandhi's idea was that if these Indians offered no violence but simply stopped cooperating with the British, stopped taking British orders, then the Brits would lose control of India. He recommended doing this through several means:

Stop working for the British. If you were employed by the British in any capacity, stop going to work, refuse to take their orders.
Stop buying British products. The British used India as a source of raw materials, which were shipped to Britain to make finished products in their factories, and the products shipped back to be sold in India. Indian manufacturing was severely discouraged - Indian factories were nationalized, those who resisted had their factories burned down. Gandhi's solution was to turn to cottage industry - make your own cloth at home, make everything you need in your own homes. Again, the reasoning was that while the Brits could (and did) burn down your textile mill or sugar refining factory, they couldn't burn down the homes of millions of Indians who were using hand looms at home to weave cloth, or any other cottage industry. Gandhi himself wore homespun cloth, used tools and utensils that he crafted himself, set up self-sufficient communes where people would manufacture everything they needed.
Stop cooperating with the British. The Brits were afraid of rebellion and had many laws in place to forestall attempts at insurrection, such as having prohibitions on meetings and gatherings. Gandhi recommended that Indians defy such laws peacefully - that they congregate in numbers to protest British rule, but offer no violence when arrested.

All of these things had an effect on the British, making it harder for them to rule India and to profit from the rule. Whether it persuaded them to leave India is a different matter, and it's hard to quantify just what role this played.

It should be mentioned that India did have a more violent independence movement as well, but much of it was retaliatory. For example, Bhagat Singh killed John Saunders, the policeman who murdered Lala Lajpat Rai. Udham Singh killed Michael O'Dwyer for his role in the Jalianwala Bagh Massacre. There were hundreds of others like them. While it made life more uncertain for Brits in India (and probably prevented them from using the brutal tactics they used against dissidents in other colonies), it doesn't seem likely that it was enough to force the Brits to leave. The scale was pretty minor.

On the other hand, it was another kind of threat of violence which played a very direct and instrumental role in the Brits leaving. When WWII started, the Indian National Congress made a deal with the Brits - the Indians would help the Brits fight the Axis, and after the war was over, the Brits would leave India. A very large Indian volunteer army fought for the Brits in WW2, mostly in southeast Asia, but also in Europe. But after the war was over, the Brits showed signs of reneging on their promise about leaving India. They sent commissions to India to discuss "limited autonomy" instead of independence, whereby India would remain part of the British Empire, but more Indians would be permitted into the administration and Indians would have more say in how their country was governed.

However, one side effect of WW2 was that it enormously strengthened the Indian Armed Forces. At the end of the war, there were about a million Indians in the Royal Indian Army, Navy and Air Force. Moreover, these were battle-hardened soldiers trained in modern warfare, veterans of the Japanese campaign in southeast Asia.

A series of events occurred in late 1945 and 1946 that caused all 3 branches of the Indian Armed Forces to mutiny against the British. Although the mutiny only consisted of refusing to take orders, and no violence was offered, the Brits realized that it was impossible to control India if the Indian Armed Forces went against them. In 1946, the British Prime Minister reported to Parliament in his annual "State of India" report that the Indian Armed Forces could no longer be relied on to prop up the Empire, that they were increasingly a nationalist force opposed to British rule.

This made India ungovernable for the British. All the talk of autonomy and limited rule ended, and in fact, the British decided to leave as quickly as possible. The timetable for independence was speeded up, British forces in India were quickly evacuated, and the British were out in August '47.

So while most of the independence movement was predominantly non-violent, in the end the threat of violence was a fairly significant factor in the British decision to leave.