The glue ‘dries’ clear and if you can do without your headphones for a while, the UV radiation in sunlight will do the trick, you just have to be patient. That is why most people use UV lights. Besides if you have the light, you can just put it aside until the next time you need it. I got a reeeeeal cheap one that is used for LOCA glue that they use for nail work to glue on decretive long fingernails but I used it for gluing LOCA adhesive for my cell phone screen protectors. Once you have it, take the battery out or insulate one side of it, and then you have it for the next time you need it. here is one like I used but there are a ton of cheap ones…(not an affiliate link)

https://www.aliexpress.com/item/1005008406384030.html

I never said it would be pretty. The other option would be to get some clay and make a mold around the break and then remove it and cut away at the break and replace the mold and UV the glue until it comes out as a resin and then take a buffer wheel to the UV Glue. Personally I would put up with the surface repair myself.

UV Glue will fix that no problems, and the glue will be stronger than the underlying plastic.