@rutwikbelsare try a USB keyboard and see if you can access your BIOS etc. with that. Your keyboards backlight working is not an indicator that your keyboard is actually working. I would suggest that you open your computer up and check the keyboard cable. Make sure it is properly seated, does not show any damage or even corrosion. If you find that all of that is properly connected, you want to consider replacing the keyboard. It is a total PITA since it is riveted, but it is possible.
Since iFixit does not yet have a specific guide for that, use something like [link|https://www.youtube.com/watch?v=kp1cwo9h1Uo|this video|new_window=true] to replace the display assembly. Slightly different model but the work is identical. Replacement keyboards are available at many online stores.
While you work on your laptop, take lots of pictures. Once you are done with it, take those pictures and create a guide for iFixit. That will help the next person that has a device with the same issues as you do. It's easy and fun to create these guides. [link|https://ifixit.com/Guide/new|https://ifixit.com/Guide/new|new_window=true] The iFixit community will appreciate this.
Repair is War on Entropy!