다가 깨다

wake up part-way through

"-다가" marks an action broken off midway, and 깨다 is waking. Almost every line is sleep being interrupted: the baby wakes up from her nap, I woke up in the middle of the night when it kept ringing, sometimes I wake in the middle. It also runs off 그러다가, "and then I woke up", which is how people end recounting a dream. One line uses it for coming round after nearly passing out. Note the first two example lines here are printed twice.