Hi. I'm not a historian (and I am Indian, so look for biases accordingly), but I believe the reason for this is that Kashmir was actually a princely state. At the time of partition in 1947, Princely states got to choose whether to remain in India or join Pakistan ( or remain independent, though this was just not a very viable plan.) Kashmir happened to be a princely state with a Hindu king but a majority-Muslim population. So Kashmir didn't have lines of partition drawn essentially because it wasn't Sir Radcliffe's duty to do so.
Now the reason Kashmir became such a hot topic today is that in 1947, it was assumed that Kashmir would join Pakistan, it being 77% Muslim. Instead, there was hesitation in joining either country. Anecdotally, I can mention that this is true even today. Kashmiris value most the unity and independence of their homeland rather than any nationalistic impulse to join either India or Pakistan. So, in 1948 Pakistan launched an attack to scare Kashmir into joining Pakistan. The maharaja Hari Singh quickly appealed to India and Lord Mountbatten agreed to military aid, provided Kashmir join India. The articles of accession were drawn up and India and Pakistan went to war in 1948, which was only ended with UN intervention.
Today, a portion of Kashmir is administered by Pakistan as Azad Kashmir ("Free Kashmir"). Ironically, just like after partition, more Muslims were left on the Indian side of the border than the Pakistani side.
The situation is also complicated by China, which built a road connecting Uighur Xinjiang to Tibet right through the Eastern portion of Ladakh, now known as the Aksai Chin region. The west of China is very mountainous and this happened to be the best route for a road. Naturally, when India eventually found out about this road, there were clashes and a war which India lost.
This history is all very complicated and like most things regarding South Asia, is spoiled by religious-nationalist tensions. As far as I have read though, this seems true. I am not very fond of writing about Kashmir as people will invariably claim I am biased. For good reason too, I am a product of the Indian education system and India is known to censor and ban books or movies if it does not fit their narrative. For a simple example, compare the map of Kashmir in an Indian atlas and one from another part of the world. The Indian maps, by law, have to show the entirety of Kashmir being Indian land, even though this is not true.
Some Indians will also claim that the result of the Sino-Indian war was not a true defeat since Chinese troops withdrew quickly after securing the Aksai Chin region and that had they continued into the Indian heartland, the superior Indian Air Force ( at the time anyway) would have put a stop to them. I think just the fact that the Indian Government could be caught so unaware and so unprepared and then be paralyzed and incapable of bold action for so long speaks volumes. It's a very heavy defeat as I can see it.
Later Pakistan also 'ceded' the trans-Karakoram tract to the Chinese in 1963. Which brings us to the situation we are in today.
You can read :
http://www.jammu-kashmir.com/documents/harisingh47.html
http://www.jammu-kashmir.com/documents/documents.html
As a counter-theory, I've wanted to read The untold story of Azad Kashmir by Christopher Snedden but have not gotten my hands on the book. If you can get it though, you should read it. It promises to show the local feelings and ideas regarding Kashmir as opposed to the Indian and Pakistani ones.