The translation isn’t so literal, so maybe that’s what’s confusing… I will talk to Kristen about it when she’s back from her time-off.
If this helps, a bit more literal translation would be something like:
I got immediately relieved once I saw my father looking well.
In addition to “one breath, one go, one sitting,” 一気 can be used for a transition/change emphasizing how fast it happened. In this case, 一気 is used as an adverb with the particle に, modifying the later part 気が抜ける. The speaker got relieved all at once when they saw his father doing well. And this emotional change happened rapidly, not slowly.