Meaning of "on the worst of days"

Hi, (excuse me for reposting this question. I've got no sufficient answer.)

The excerpt below is from a book by David Sedaris.

Would you explain what "On the worst of days" means? Is it like, on the worst days? or on the worst part of the sessions, which probably took place at the end of each session?

(He, David, has a lisp issue here. Agent Samson is actually a speech therapist. It's being used metaphorically.)
My sessions varied from week to week. Sometimes I'd spend the half hour parroting whatever Agent Samson had to say. We'd occasionally pass the time examining charts on tongue position or reading childish s-laden texts recounting the adventures or seals or settlers named Sassy or Samuel. On the worst of days she'd haul out a tape recorder and show me just how much progress I was failing to make.

Thank you,
M
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Sorry, but one more thing.
What is the difference, in this context, between "I'd spend half hour" and "I'd spend half hour"?