There may be exceptions to this, but usually a visa only has to be valid when you enter a country, not when you leave it. When you enter a country, your passport gets stamped "admitted until xx/xx/xx". Your "admitted until" date can be after the expiration of your visa. As long as you don't overstay your "admitted until" date, you're fine.

It's not clear to me from the original post whether the poster is already in-country, or whether the date that's about to pass is the visa expiration or the "admitted until" date. If you're already in-country and the "admitted until" date is about to pass, I'd go to immigration and explain the situation before the "admitted until" date passes. If you're not in-country, a cancelled flight won't give you any leeway as to a visa expiration date for entering.

Disclaimer: I'm not an immigration lawyer anywhere, certainly not in Korea. I'm sure the above is usually the case, but you should verify that it applies in Korea in particular before you rely on it.