Bookmark the following places for sure -
1. The cafe at Korean Cultural Centre in Lajpat Nagar
2. Kori's - Safdarjung & MKT
3. Busan Restaurant - MKT
You can even pick up authentic instant ramyeon packets from INA Market if you're in the mood for snacking.
Gung The Palace in Green Park and Shimtur in Paharganj are my fave go-to places for this!
Busan korean restaurant in Majnu ka Tila and AIM cafe in Korean Cultural Center, both are really good restaurants serving korean food. AIM cafe in Korean Cultural Center offers you genuine korean dishes and teas.