That’s strange, it looks like your USB port is missing a structural piece that holds the contact points in place. Look at the cable you charge the calculator with, does it have a black piece stuck in it?
Here’s an example of what mine (top) looks like compared to yours (bottom).
Are you familiar with soldering or know someone that can solder? I’ll ask around but I’m pretty sure the only way to fix this is going to require replacing the USB port. You can take a look a the teardown guide so you kind of know what you’re getting into.
Meanwhile, the only way to charge your calculator is to remove the battery and put it in a friend’s calculator, then charge it through their calculator. It should last you a month or so on a full charge if you keep your brightness low (press [2nd] then hold down the down arrow to reduce the brightness).
I would check if your calculator is under warranty still, perhaps TI can replace the calculator for you. If worst comes to worst and you have to buy a new calculator, honestly a TI-83 Plus for $20 on Ebay will get you by Algebra 2. If you have a little extra money I would recommend the TI-84 Plus for about $50-60. Don’t get a TI-84 Plus C Silver Edition! Although it has a color screen, it’s the slowest calculator made by TI and you wont have a good experience with it at all.