V/A+다는 것 is a short form of the indirect quotation pattern V/A+다고 하는 것, which means "the thing that is called + V/A" and it is used to express what you think the definition or the nature of a certain action/state/verb. 영국에는 역사적 명소가 많다는 것을 안다. I know that it is said that there are many famous places in England. (knowledge by indirect experience) 영국에는 역사적 명소가 많는 것을 안다(X) 영국에는 역사적 명소가 많은 것을 안다(O) I know that there are many famous places in England.(knowledge by direct experience) 많는 is wrong conjugation because 많다 is an adjective. You are correct. 이미 끝나 버렸단 걸 알까? = 이미 끝나 버렸다는 것을 알까?