To the wonderful guest who asked me for an explanation. I wish you'd left an account so I could PM you, but I guess the question would've come up anyway.
The O/S was meant to be confusing, and actually illicits puzzling out by the reader. In fact, while I was in the process of writing it, there was a moment where I thought I became too clear.
But anyway, each paragraph is alternating between Magnus and Jace with Alec. And because Jace has been incredibly insensitive to Alec's feelings and actually just focused on his relationship with Clary, Alec has been drinking himself to oblivion to the point where he drinks too much. Which is when he is hospitalized.
Magnus has been his "drinking partner", but he does not do it excessively like Alec does. While Magnus has his sober and drunk moments, Alec spends all of his days drunk, too hurt by the way Jace blatantly ignores his feelings.
The hospital pumped the alcohol out of his body, and that was his first sober moment in a while. Without a substance clouding his emotions and judgment, he was finally able to see how he actually felt, and for the first time, he saw Magnus.
I really hope this helped, and if you have any further questions, please let me know.