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Why was the Kashmir issue not completely solved at the end of Bangladesh Liberation War of 1971?
A 1948 UN resolution (Wikipedia summary) on Kashmir says there should be a plebiscite or referendum regarding the determination of the fate of Kashmir, and this fate would be determined by none other than the Kashmiri population. 1971 war had nothing to do with this said referendum.
Also, India couldn't open a new front in Kashmir because of the risk of China's involvement which the USSR kept in check. Remember, China claimed a part of Kashmir called Aksai Chin, and the Saksgram Vally was delivered to China by Pakistan through a land swap agreement. Thereby China and Pakistan became allies after 1962.
There was also a complex equation in progress regarding superpowers. The USA did not favor the USSR having too much influence in the region.
Why could the issue not be solved and made a two-party affair?
Kashmir issue is a two-party affair according to India. Pakistan wants it to make a multi-party affair which India opposes.
I wrote an answer in politics.SE regarding this.
Why not partition Kashmir, or give it whole to one party or something like a final solution?
Pakistan wants a referendum, and no one, not even a superpower, can exclude Pakistan from this dispute.