Passport from Republic of China traveling to Taiwan with layover in Seoul

According to Korea Immigration Service's website, I see two possibility.

Because the journey starts from New York, U.S., if she has U.S. visa, this applies...

Tourists in transit to Third Countries

1. Eligible Applicants

Citizens of all countries except Cuba, Macedonia, Syria, Sudan, and Iran.

2. Requirements

Persons who hold a visa (re-entry permit, permanent residency, etc.) to enter the United States, Canada, Australia, or New Zealand (hereinafter referred to as “the four advanced countries”) and

1) who go to one of the four advanced countries in transit through Korea; or

2) who stay in one of the four advanced countries and take a direct flight from the country to Korea to go to a country of nationality or a third country.

Please be noted that:

• If you are a U.S. visa holder who transit through Korea and Saipan to reach the U.S. as the final destination, you are permitted to enter Korea without a visa. (However, if you depart from the U.S. and enter Korea via Saipan, you are not granted no-visa entry to Korea.)

• If you are a Chinese group tourist holding a group-tourist visa to Japan and meet certain requirements (e.g. using a travel agency that only attracts Chinese group tourists), you are permitted to enter Korea without a visa.

• If you hold an e-visa to Australia and a visa label is attached to your passport, you are permitted to enter Korea without a visa. However, if the label is not on your passport, you are granted no-visa entry only when you depart from Australia and transit through Korea.

3. Condition of Permission

You have a flight ticket that is scheduled to depart from Korea within 30 days, and have never violated laws (e.g. overstay, etc.) after entering the United States, Canada, Australia, or New Zealand.

Or if not...

Visa Free Entry for foreign transfer passengers

1. Eligible Applicants

Besides the passengers for transit mentioned above, those transferring through the Incheon Airport and taking part in the Transit Tourism Program on their way to a third country or the country of their nationality (hereinafter referred to as "general transfer passengers")

2. Condition of Permission

General Transfer Passengers: Possessing a transfer ticket and staying in the Seoul Capital Area for up to 72 hours

3. Eligibility and Period for Stay

General Transfer Passengers: Tourist/Transit (B-2), 72 hours

Note that Seoul Capital Area is "Seoul, Gyeonggi-do, Incheon". In this case, you should not go beyond the red zone below, and you'll be fine. (Staying inside the red zone is fine.)

If you are not transferring through Incheon, it's not applicable, though it's low chance you're taking a flight through Gimpo.

Image source: Dmthoth/ASDFGH, CC BY SA 3.0/GFDL

PS: I have no idea what George Y. is mentioning about the national/household ID number stuff means in this case.