QUEENS:

Lee, Hyori, BOA, SES, and SNSD.

KINGS:
For me, it's between DBSK, H.O.T, Rain, BIGBANG and Shinhwa.
The Hallyu wave was started in 1992 by Seo Taiji and popularized in 1996 by H.O.T, and they also promoted K-Pop in China. Remember Candy? Yeah, that song was a must-cover for idols.
Rain became popular, too, not only in Korea but also in various countries. So I guess he's one of the royalties.
DBSK. This is pretty much self-explanatory. If you're a real K-Pop fan, you'd know why these guys (the five of them) are kings.
Last but not the least, Shinhwa, the longest running group with no member line-up changes. They started in 1998, although their debut wasn't successful (I think it's because of their song Eusha Eusha, which is about summer, and then a storm happened. Yeah, you figure out the rest.) they became one of the most respected groups out there.
Actually, you can't really pinpoint who's the real king and queen. It depends on the generation.
For me, there's a certain queen/s and king/s in very era.
During the 90's, it's between H.O.T and Sechskies (have you guise watch Reply 1997? You'd know who they are, and how fans at that time acted.)
G.O.D became popular too, but the two former groups were more popular.
In 2000's, it's Rain, DBSK, and Shinhwa.
BIG BANG became popular, too, and their fandom is pretty large.
Right now, I think the current kings are EXO and BAP. I don't know much about BTOB and JJ Project. Sorry.
And well, I'm not really following the new groups. I'm more inclined to the older groups ^^

KINGS: DBSK. Need not to explain, they are the GODSSSS and KINGSSSS of Kpop =)
QUEEN: BoA. I loayk her. <3 And she's perfect I swearrr.

KING S= BIG BANG
because theyre handsome and kind :)
QUEENS= 2 NE 1
because theyre pretty and kind :)

Lee HyoRi = Queen of K-Pop
ShinHwa or TVXQ = Kings of K-Pop
come to think of it, I guess ShinHwa are more the legends than kings

SuperJunior are the kings = because in my opinion they are simply amazing and flawless and down to earth and kind and every loving word there is in the world to describe them! and they actually have talent!!!!!!
queens? = i don't know, it's between BoA and Lee Hyori in my opinion?

BoA - a real talent who had debuted at a young age if 13 . Hits high notes with those ultra powerful dance moves.
DBSK - is there a need for a reason ? They hit those beautiful notes effortlessly (*cough* Changmin screams *cough*) and their dance is like so powerful and they can be adorable yet charismatic at the same time . Good actors too ~