Quoted from Da-Shaker:Is there a rivet or something similar to replace the metal around the holes?
Yes. But....
Delicate work and you have to have the right tools including a really good soldering pencil and high quality solder. This video....
...is not exactly as you describe the condition of your board, but should give you an idea. If you've never done it before you should get some help from someone who has the tools. Your Weller gun will not do this job.
OTOH, Vid may have some better suggestions. He always does.
///Rich