Nice hotel choice.

For Korean BBQ, I've really enjoyed eating at BonGa. Or as they spell it, Bornga.

http://bornga.co.kr/bonga/index.asp

Take a look at the pictures on http://bornga.co.kr/bonga/menu_woo3.asp and tell me if that doesn't get your mouth watering! They even tell you how to eat it. Dip in sauce, place in vegetable wrap and some other sides and enjoy! This is what they are known for.

Also, http://bornga.co.kr/bonga/menu_chadol.asp is the second dish they are known for. The bean stew along with the mixed rice, or bibimbap.

I only know of one location in Shin Nonhyeon area but apparently there are multiple locations through out Seoul. I was told that the one in Shin Nonhyeon area is open 24 hours. First floor has tables but the second floor, you sit on the ground. Be prepared for a wait if you show up during normal dining times. I will try and get you more specific instructions on how to find it. But the front of the restaurant is the one that shows up on the first link I posted above.

http://bornga.co.kr/bonga/store_domestic.asp

This page has the list of locations in Seoul. I apologize I can't assist further in finding the one closest to your hotel or the nicest one to visit. But perhaps someone in this forum can help with it.