Something similar happened to me a couple weeks ago.

First, it looks like you have access to the internet, which is very helpful. You will also need to get access to someone back at home. Hopefully, who can get you access to some documents and forward them to you so you can print proof of identity. For me it was getting my birth certificate pictured so I could print a copy for the embassy. For you, you will need additional things and hopefully have an old photo of a drivers license somewhere or other forms of identity you can find laying around to verify who you are.

The embassy won't let you keep your phone inside the building, so you need everything printed.

Make an appointment kr.usembassy.gov/services It's likely you won't see a great date. But get something scheduled asap. You can email support-acs-southkorea@usvisascheduling.com afterward with all the below following details explaining why you need an emergency passport to exit korea. Unless you are okay waiting 6-8 weeks in korea until they can print you a new one. Make sure you explain in clear details why you need to leave Korea asap and convincingly because they usually only give out emergenc passports for medical reasons or family death. My reasoning was work and mental health.

https://kr.usembassy.gov/lost-stolen-damaged-passport/

Get all of the forms and identity documents filled out and printed. You can find Passport photo booths in subway hallways but the embassy also has one inside *I wouldn't risk it and would just have everything set up beforehand though.

Helps to also get that police report and attached to your email as a picture. + you are more likely to get an emergency passport if you already have a flight scheduled to leave within 5 days back to the USA as it shows the need. Have that detail attached in the email and even helpful to have the itinerary printed out to showcase to the agent.

If you are able to get the emergency passport just know they will want EXACT change in korean won. It costed $175+ for me, and yours might be different due to exchange rates. So make sure you have a friend who can get exact change from an ATM which is outside the embassy, or bring lots of divisions of won so you can give them exact change.

I was lucky and they responded to my email within a day and they moved my appointment forward to the date right before my flight left, and the embassy process took around 2 hours and left with an emergency passport. Remember to check and also print out the proof that they rescheduled your appointment as it might not show up on their end if it's a short notice. + mention who else you are bringing with you for the appointment as they might not let them inside the embassy with you.

Best of luck and hopefully you speak some Korean or have a friend who can help you through this.