Clean out the headphone jack with a qtip, may have to remove some cotton, and using 90% isopropyl alcohol
UPDATED ANSWER
Ok, so since that didn't take, the easiest option for you would be to replace the entire motherboard. Yes. You could get away with desoldering the broken headphone jack, and soldering a new one on. However, and this would mostly be with the desoldering to be the issue, the knowledge of doing so would be taxing, without the knowledge of doing so. Also, I looked for specifically that part, and didn't find it an easy search. Here's a list of the part you'll need off eBay, and I'll list a guide on how to do so
https://www.google.com/search?q=gameboy+...
Game Boy Advance Logic Board Replacement
crwdns2934105:0crwdne2934105:0
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Do you know if it was playing sound trough the speakers before you got it? Or did you just discover this problem?
crwdns2934271:0crwdnd2934271:0 Alex Niculescu crwdne2934271:0
It was playing sound when I got it
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