Choose a research topic in which you are interested in conducting research. Check the IIT's official website to see if any faculty members are conducting research in your field of interest. After that, send them a personal email with your resume and cover letter attached. You simply need to be honest, demonstrate your true dedication, and explain the importance of the internship to your future. Obtain a reference letter from your HOD and send it as well. If they are interested in hiring you as an intern, they will definitely respond to your email. The same procedure can be followed for ISI Kolkata.
If you know anyone at IIT or ISI, it will be very easy to obtain an internship offer from them.
One of the best ways would be to look up the research areas of the professors and then decide which professor’s area matches your area of interest.
After doing that, you can try going through some of the papers that the professor had published and then mail them stating that you are interested in their area of research. Read the papers thoroughly and appreciated some points in your mail and mention that you wish to work with them over the winter. If you have any research experience, then don’t forget to mention that too.