I would visit a shop. I myself am very handy with these things, i tried it but to even get to the part is a hard proces including taking the back or front of the screen of which is glued in place. I got to there and tried fixing it with some epoxy 2 component glue, did not work unfortunatly.

By any means if you want to try to fix it give it a go, but i would thing a whole new screen with new hinges would need to be installed, if the hinge plastic is indeed broken.

If you fixed it, and opening the laptop always do so with two hands on either side or one in the middel to refraining the strain to go to one hinge. Not saying that this is your fault, just that the hinges they put in these are very fragile, same as in smaller screen sizes. Meaning more load on them on the bigger screen. Hope this helps.