What do you call something you've said so often you don't hear yourself speaking?

For a fascinating discussion of how repetition drains meaning from words, see Haiman 1990 “Ritualization and the development of language” (google books; photocopy). The process you are describing is I think a form of emancipation (§1.1) and habituation (§1.2). You are observing a decline in response to a stimulus that has become familiar through repeated exposure.

The thing you are saying is Ritual Speech, a kind of cliché.

I've given this speech so many times it's become ritual speech.

I've given this speech so many times it's become cliché.

You don't remember the speech because, to your brain, it is a single chunk: the individual words no longer have meaning for you. This doesn't mean that they don't have their proper effect. Du Bois 1987 suggests that ritual speech doesn't have intention. There is no difference between using words to control an Azande poison oracle, using your hands to control a microwave oven, or giving a ritual speech to get to the next step in a treatment.

But it would be interesting to record yourself giving the speech. Haiman describes different signals, such as stress, sing-song intonation, speed, etc. that betray ritualization. Your patients might be hearing a message that you don't mean to convey.