Hi @snufflekin ,
Is it that the hinges are broken or that the posts the the hinges screw into in the palmrest assembly have broken?
If it is the posts, then you need to replace the palmrest assembly.
Here’s a link to the [https://topics-cdn.dell.com/pdf/inspiron-15-5577-gaming-laptop_service-manual_en-us.pdf|service manual] for your laptop, taken from this [https://www.dell.com/support/home/us/en/04/product-support/product/inspiron-15-5577-gaming-laptop/manuals|webpage].
Scroll to p.43 to view the necessary pre-requisite steps and then the procedure to remove/replace the palmrest assembly.
Replacements are available online, just search for ''Dell Inspiron 5577 palmrest assembly'' '''or''' ''Vf544 0VF544'' (part number - see bottom of assembly to confirm) to find suppliers