Kashmir or Sikkim?

We're looking for something on a par with Everest or Annapurna but in India. We're looking for a challenge over 5,000m. We're drawn to Sikkim and Kashmir, but what would you suggest?

Keeping all your criteria in mind, I'd actually suggest Ladakh. It offers grand landscapes, a dose of Buddhist culture, and is somewhat close to travel standards in Nepal and should live up to your expectations. While Kashmir is beautiful as well (more alpine and lush) it's a bit more complex with its permits and other things, especially for foreigners, because of it being slightly politically volatile. The better alternative in that case would be Sikkim, but my first recommendation would be Ladakh.