Possibly, "I do know that" can in fact only be used, when, you are answering the question of whether or not you know the issue at hand (or your knowledge has been called in to question, and you are answering that challenge). Let's say "out of the blue" you wanted to state that "you know that" -- and you wanted an emphatic version.
PC Magazine: Want to See Lyrics on YouTube Music? You Might Need a Subscription
Original Story 2/9: YouTube Music now requires users to have a Premium subscription to view song lyrics. Google hasn’t confirmed the change yet, but users on the free tier are posting screenshots of ...
Want to See Lyrics on YouTube Music? You Might Need a Subscription
Recently one of my friends told me that there is distinct difference between 'know of something' and 'know about something' expressions. 'know of' is used when you have personal experience with wha...
"Know about" vs. "know of" - English Language & Usage Stack Exchange
Examples: I know (about) this difficulty/problem. I know (about) Engineering. In the first sentence, it seems to me that "to know" expresses that the speaker experienced the problem/difficulty before while "to know about" only expresses that the speaker has heard or read about it.
to know vs to know about - English Language & Usage Stack Exchange
Why do you think that He doesn't know him from his schooldays means that he does know him? It would only have that sense if you added something like In fact, he first met him at university.
Know in (1) refers to the clause that comes right before it, so there's no pronoun necessary -- it's essentially a transform of I know it's your job. In (2), however, the object of know is not indicated, as you point out, so something must be provided.