“More than reflect”, what’s the subject here?

This is, you are right, a roundabout way of saying that she went to the place where he lived and didn’t see him. But this is a very strong (emotional) way to say that. Not only did s/he not see him: s/he did nothing else. So s/he spent the entire time thinking to her/himself (reflecting) “I’m not seeing him.”. You might think s/he would have found other thing to do with her/his empty time. But no, s/he did nothing but eat, drink breathe and fret about the fact that she was not seeing him. It is a kind of rhetorical hyperbole.

The verb I reflect can mean the same as I think or I consider. So, like these, it can function as the main verb in reported speech (or, in this case, thought). This is clear in the definition in the Cambridge English dictionary, which provides the following example:

She reflected that this was probably be the last time she would ever see him.

So in your passage it is fine to write

I did nothing but reflect that I ...