I've worked on this list of the rulers of Korea. Hope it can be used, and if not, then at east be a help to make a better list
Western Temple names are used for the kings.
Era names used for the emperors.
The last king and the first emperor are the same person though.
The Japanese Protectorate under the Korean Empire can probably just be fused and be only the empirer of Korea, untill the Japanese anexed Korea, in 1910.
This list should work for coins and banknotes.
Kingdom of Goryeo, Wang Dynasty
- Sukjong (1095-1105)
- Yejong (1105-1122)
Kingdom of Joseon, Yi Dynasty
- Sejong the Great (1418-1450)
- Injo (1623-1649)
- Hyojong (1649-1659)
- Hyeonjong (1659-1674)
- Sukjong (1674-1720)
- Gyeongjong (1720-1724)
- Yeongjo (1724-1776)
- Jeongjo (1776-1800)
- Sunjo (1800-1834)
- Heonjong (1834-1849)
- Cheoljong (1849-1863)
- Gojong (1863-1897)
Empire of Korea, Yi Dynasty
- Gwangmu (1897-1905)
Empire of Korea, Yi Dynasty, Japanese Protectorate
- Gwangmu (1905-1907)
- Yunghui (1907-1910)
Japanese Protectorate
- Meiji (1910-1912)
- Taisho (1912-1926)
- Showa (1926-1945)
Sources:
https://www.metmuseum.org/toah/hd/koru/hd_koru.htm
https://en.wikipedia.org/wiki/List_of_monarchs_of_Korea
https://en.wikipedia.org/wiki/History_of_Korea
Video about the Kingdom of Korea, made by UsefulCharts:
https://www.youtube.com/watch?v=gdP-U4sEcD4&list=TLPQMDIwNTIwMjG9Bl4FHIkC5Q&index=14
https://en.wikipedia.org/wiki/List_of_emperors_of_Japan
https://en.wikipedia.org/wiki/Korea_under_Japanese_rule
https://en.wikipedia.org/wiki/Korean_Empire
https://en.wikipedia.org/wiki/Japan%E2%80%93Korea_Treaty_of_1905