Plato and the definition of knowledge

epistemology (from Greek episteme, ‘knowledge’, and logos, ‘explanation’), the study of the nature of knowledge and justification; specifically, the study of (a) the defining features, (b) the substantive conditions or sources, and (c) the limits of knowledge and justification.  The latter three categories are represented by traditional philosophical controversy over the analysis of knowledge and justification, the sources of knowledge and justification (e.g., rationalism versus empiricism), and the viability of skepticism about knowledge and justification.

[Excerpted from The Cambridge Dictionary of Philosophy.]

Shannon’s (over-?)simplified retelling of the dialogue:

Soc.  What is knowledge?
Theaet.  True judgment?
Soc.  Is the distinction between true and false judgments helpful?  How are false judgments possible?  Can I know (or judge or believe) something that is not any more than I can see or hear something that is not?  [Socrates is here concerned with the object of our knowledge.  Since knowledge must evidently have an object, how can that object not be?  On these grounds, Socrates concludes that false judgment is, strictly speaking, impossible.  He therefore rejects Th’s proposed definition.]
Soc.  So, what’s knowledge?
Theaet.  True belief?
Soc.  But (as the legal profession teaches us), we can believe what is true for the wrong reasons.  Surely, we don’t want to call this knowledge?
Theaet.  So, what if we say that knowledge is true belief “with an account”?
Soc.  “With an account” must mean one of three things:  (1) A mere verbal explanation of something, (2) an enumeration of a thing’s elements, or (3) being able to get hold of the difference distinguishing a thing from all others.  The first two can’t be right because they’re just rote learning, and don’t show a real understanding of the thing in question.  So, it would seem that we want the last notion of “account.”  
The third way seems to show us the difference between an account of a thing and a correct account of a thing.  So, the correct account of a thing, is just the catalogue of that thing’s qualities, plus whatever makes that thing distinct from others.  But, if this is right, then when I was merely enumerating the thing’s qualities, I didn’t yet know about those qualities that made it really distinct.  So, how could I have been enumerating that thing’s qualities at all?  It seems that the only notion of a particular thing is necessarily a correct notion of a particular thing.  But this is circular.  
So, maybe what is important about the idea of an “account” is not the content of the account, but the process – how we get to know the differentness of a thing.  But, then, we have another circle:  knowledge is correct belief together with knowledge of differentness.  That seems to be all that “adding an account” does for us.  (And it’s not enough to be helpful.)
Back to the old drawing board…

Can we do better than Socrates and Theaetetus?  What is knowledge?  What do we mean when we say that S knows that P?