An independent Kashmir (not including Jammu or Ladakh) would probably be a deeply authoritarian state, marred by ethnic cleansing of minorities and general isolation from its much larger southern neighbour. It would even have a China-esque relationship with Pakistan and vitriolic hatred for, and general isolation from, India, strengthening the similarities with North Korea.
Here’s a scenario on that note.
Jammu and Kashmir successfully achieves independence from both India and Pakistan, attempting to be a monarchist and South Asian version of Switzerland. However, its independence is met with Hindu and Muslim riots. Kashmiri nationalism grows among the Kashmiris, while pro-India sentiments grow among the Hindus. Eventually, after the king dies, the Hindu ultranationalist RSS takes control of Jammu, Kashmiri nationalists take control of Kashmir, and Ladakh restores its monarchy but is ignored by just about everyone. In both Jammu and Kashmir, minorities flee riots and suspicion, homogenizing both lands. Jammu is integrated by India, which promptly forces it to become part of the state of Himachal Pradesh in an attempt to reduce the power of the RSS. Kashmiri nationalism, however, has grown to the extent that it doesn’t desire to become part of Pakistan, instead becoming ruled by Muslim Kashmiri nationalists. In reaction to the ethnic cleansing of Kashmiri Hindus, India invades Kashmir, placing a puppet government in charge.
This puppet government proves unpopular, further radicalizing Kashmiri nationalists. However, any Kashmiri nationalist coup is frustrated by Indian intervention, and eventually when India gets nukes, it is forced to toe the line. This all changes when Pakistan gets nukes, acting as a deterrent against India. A Pakistan-supported Islamist and Kashmiri nationalist coup successfully takes over Kashmir, and India is kept from intervening by Pakistani nukes pointed at Amritsar, Jammu, and other major border cities. Instead, it attempts to cut off Kashmir, resulting in a famine which is only rectified by establishing ties with Pakistan. Despite finding the radicalism of the Kashmiri government a bit too much, Pakistan nevertheless continues to aid it if only to keep India from getting it. Therefore, Kashmir, despite pariah status, continues to live on under a totalitarian Islamist and ultranationalist government.