I didn't understand the question at all. Why shouldn't the Holy One, blessed be He, be able to know that He knows? Is there some paradox here that I'm missing?

Just as I, or anyone else, can't determine that there isn't something we don't know. After all, if we don't know it, then maybe we don't know that it exists.

I still don't understand the difficulty. I'm not, God forbid, trying to be annoying, but I just don't understand what is hard about the question.
The question was: how does the Holy One, blessed be He, know that He knows everything? I don't understand the problem—if He knows everything, then He knows that too. In addition, afterward the questioner adds that there may be things He does not know… so if that's really the case, then why get into the whole hair-splitting point that "He also doesn't know that He doesn't know them"? Just say there really are things He doesn't know. What I understood from you is that if, for the Holy One, blessed be He, there may be things He doesn't know, and indeed there are such things, then in addition one can also claim (under that assumption, though it doesn't necessarily follow) that there are things He doesn't know that He doesn't know. In other words, the Holy One, blessed be He, is confused, and the fact that He tells us He does know everything is only because He doesn't know about what He doesn't know. In addition, this also assumes that everything He doesn't know, He also doesn't know that He doesn't know it, and that there aren't things He knows that He doesn't know.
So it comes out that there are two levels of not knowing:
1. I know that I don't know
2. I don't even know that I don't know

In short, I don't understand where the difficulty is. If I were assuming that the Holy One, blessed be He, knows everything and from that arriving at a contradiction, fine. But to say that the Holy One, blessed be He, told us that He knows everything, and it may be that He doesn't know everything, and therefore it may be that He doesn't know that He doesn't know something—that's begging the question (with some unnecessary cleverness in the middle), no?

Side note: it's interesting to think—if there really are things that we can't know that we don't know because we have no access to them whatsoever, then how did we determine in the first place that there are things in that category? What's the indication of that?

Amir, it's simple.
God thinks He knows everything. But how can that be known with certainty? After all, the fact that He knows ten things still doesn't guarantee that there isn't an eleventh thing whose very existence He doesn't know. That requires an additional level of awareness.
Think of looking at a soccer field and clearly seeing all the balls on it.
Where do you get the confidence that maybe there isn't an invisible ball whose existence you don't know about?
You have to distinguish between two types of awareness:
A) I look at the whole field and see all of it.
Clearly, that clause does not guarantee that I see everything.
Successfully seeing every point in space does not guarantee certainty that I actually see what *is* at every point in space.
B) Knowing that there isn't something that exists on the field that I do not see.
So too with God: although the entire universe is spread out before Him, how does He know that there isn't some object in space (which is entirely spread out before Him, as stated) that He is unable to see, even though He is looking at its location? That requires awareness at the level of clause B.
Therefore the Rabbi answered that apparently He has some particular kind of awareness (which is not clear to us, like sight is not clear to a blind person) that allows Him to know that there is not something that He does not know.

Exactly. Well said.

If so, then it really is simple, and that is indeed how I understood it after a bit of clarification from the Rabbi. My claim afterward was that this is simply begging the question and not a real question. The initial assumption you're starting from is: it may be that there are things that the Holy One, blessed be He, doesn't know that He doesn't know about. From there you conclude that when the Holy One, blessed be He, tells us He knows everything, it may be that He's missing things (because He doesn't know about them). Therefore the Holy One, blessed be He, does not necessarily know everything. I asked why assume that at all, rather than assuming, for example, that He knows everything (after all, if He created everything, it's reasonable that He would know everything). Still, I will qualify that by saying I agree that the question is legitimate if you are willing to adopt only the assumption that He merely thinks He knows everything, and not that He actually knows everything.
From there, you really can explain in two ways how such a statement could make sense:
1. Either He knows everything
2. Or He doesn't know everything, but He doesn't know what He doesn't know, and so He is nevertheless confused and tells us that He knows everything.

P.S. Personally, I don't think that the Holy One, blessed be He, actually knows everything, but I went along with the question.