Hi. This is a common problem with Kindle. I am still "amazed" that amazon don't do a recall for that issue. I guess that the size of a company is not an indicator about amateur they are.
You can bring it at an electronic repair shop (one that solder thing not the crap they call repair shop now that are only part swaper. Even if they never handled one yet, there is no shortage of how to on youtube.com about making the repair.
I would need to know your location to be able to find you a repair shop that could solder it back.
Mailing is very expensive these days. Instead, look into those two local business. The first link can even go fix at your home:
Weird as those are very similar, especially soldering the port. Hopefully the second will do it.
Take all the time you need.