I'm web security professional and I've found major security issue with one of the leading Nepali news website and I've tried to notify them via email but neither they respond to that email nor solved the issue.
In this situation what I can do?

Most of the websites that i've seen have some kind of vulnerabilities anyway.

I'd , if i were you :

1. Check the Registrant for the domain name. Usually, they put their email address there. Notify them.
2. Check the website - Contact page, and notify on a more generic email. (admin@website.com , support@organisation.com)

Remember. They might have 'asked' someone else to build the website for them, and they might simply not have any clue to proceed to patch their website against the vulnerability. In such case, if you know the exploit and the solution .. it would be awesome if you help them in that.

For any reasons, if they dont take any action. You could as well put a banner on their website. Its always better a Nepali hacker/security professional let knows of a problem in a Nepali website/Server than a Turkish one ; )