얼마나 A/V + ㄴ/은/는지 exaggerates the magnitude of a feeling or action. If used with a verb it usually needs an adverb, e.g. 많이, 못, 열심히, 오래...
시험을 위해 얼마나 열심히 공부를 했는지 D로 겨우 통과했어요!
I can't even tell how hard I studied for the test... and then I barely passed with a D!
가게 밖에 줄이 너무 겨서 얼마나 오래 기다렸는지 몰라요.
The queue outside the store was so long that I couldn't even tell you how long I was waiting.
실수를 해서 제가 얼마나 미안한지 몰라요.
I can't tell you how sorry I am for my mistake.
얼마나 화가 나는지 얼굴이 빨갛게 변하고 고함을 질렀어요.
I was so inexplicably angry, my face turned red and I was shrieking.
딸의 음악 공연을 구경하고 얼마나 깊은 인상을 받았는지 몰라요.
I was so deeply impressed when I watched your daughter's music recital.
이 개그맨이 얼마나 재밌는지 저는 웃음을 멈출 수가 없었어요.
I can't even tell you how funny that comedian was -- I just could not stop laughing.
회식을 했을 때 음식을 얼마나 많이 먹었는지 그 다음날도 배가 아팠어요.
When I went to the company dinner, I ate an unbelievable amount of food -- my stomach was still hurting the next day.
대구 교통이 얼마나 복잡한지 몰라요.
I can't even explain how congested the traffic is in Daegu.
요즘 리야드의 날씨가 얼마나 더운지 죽을 지경이에요.
I can barely describe how hot it is in Riyadh these days -- I am on the verge of dying.
영화를 보고 얼마나 감동한지 몰라요.
I can't tell you how moving the movie was.
산꼭대기에서 경치가 얼마나 아름다운지 몰라요.
The views from the top of the mountain are indescribably beautiful.
이 드라마가 얼마나 재미있는지 몰라요.
This drama is soooooo exciting!
신칸센 기차가 얼마나 빠른지 도쿄에서 오사카까지 3시간 안에 갈 수 있어요.
The bullet train is soooo fast it's remarkable -- you can go from Tokyo to Osaka in under three hours.