You cannot fix this screen, it is broken beyond repair.

Using the MacBook Air 2013 with an external screen is a very good idea, it is limited to 2560 x 1600 Pixels. Although, replacing the screen on an MBA 2013-2017 is super easy. There is a great detailed video available here (and the Youtuber also lists what tools and parts are needed).

Screens are sold on ebay for about 40-60 USD.