What kind of USB adapter do I need for my broken headphone jack?

Make sure you get a 4-pole TRRS converter; otherwise, your microphone won't work. Look for one designed specifically for TRRS connectors, not just any DAC.

You can't just plug analog headphones directly into a USB port since they're digital. If you're set on using USB, you’ll need a converter of some kind.

I recommend searching for a "USB soundcard". I personally use this one: https://au.creative.com/p/sound-cards/sound-blaster-play-3#buy-menu. It has both the TRRS jack for headsets with a mic and an extra port for a standalone mic if your headset comes with two plugs.

What you're looking for is called a USB DAC (Digital-to-Analog Converter). Searching for that on Amazon will show you a variety of options, including some simple USB to Aux adapters. I’m not really an audiophile, but Fosi is a trusted brand, and this one seems pretty decent: https://www.amazon.com/Fosi-Audio-K5-Pro-Headphone/dp/B0BFHFSX4C

What type of connector does your headset use? If it's a big 1/4" jack, there’s an adapter to convert it to 3.5mm. If it has separate audio and mic connections, you’ll need a specific adapter to connect both to the USB DAC you plan to buy. Let me know, and I can help you find a suitable link on Amazon!

Try looking for a USB DAC/amp. These usually range from about $20 to $100 based on how high-end you want to go. Your headset should work fine since they mostly use a standard 3.5mm jack unless it's some unique gaming headset with its own connector.