"Daniel fell asleep sitting on the sofa."
Translation:Daniel a adormit stând pe canapea.
DL should however also accept the gerund of "a ședea", since by definition it is much closer to "to sit" than "a sta". Not sure if "a ședea" isn't a regionalism though.
"A sta" literally means "to stay", irrespective of whether you're sitting, standing, leaning or otherwise. You just have to be stationary (even when relative to other forms of movement: answering a telephone call, you could technically answer "Stau în mașină/tren/avion”, even if the vehicle is moving (as long as you're not driving, since you're not moving relative to the vehicle)).