Everything hinges on how you understand "knowing."
First off, you're going to need to decide whether there's a distinction between episteme, techne, and phronesis (or if you prefer Gilbert Ryle's "know-how" and "know-that"). It seems like if we accept this, then there's most definitely a distinction between "knowing how to kick a ball" (techne /know-how) and "knowing that you know how to kick a ball" (episteme concerning facts in the world / know that).
Presumably, the intent in axiom relates only to episteme / know-that. But even then, I don't think the axiom follows easily. In part, because I take these claims to be reflective judgments about the world. Using the "I know that ..." or "I believe that ..." structure, I would say there's a few different types of cases we can suggest as counter examples:
Type 1 - Failure to realize knowledge of all members set means knowledge of the whole set.
If I know separately that 2,3,5,7,11,13,17,19,23,29,31,37,41,43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, and 101 are each prime numbers AND that there are no intervening prime numbers. Then I can draw the implication that I know all prime numbers less than 100. If I do draw this implication, then I know that "I know all the prime numbers less than 100" but until I do, it's not clear that I know this.
Similarly, if I know that Elizabeth II is the current Queen of England and George VI , Edward VIII, George V, Edward VII, and Victoria precede her, then I do in fact know the names of all British monarchs in the 20th century. But do I know that I know that? Here, I would need to know additional information about them to infer that.
-- if you don't take knowing to be an action then it's not clear if this will convince.
Type 2 - failure to make a comparison
If I know that Hydrogen is lighter than Helium and that Helium is lighter than Gold, I still need to connect these bits of knowledge to know that Hydrogen is lighter than Gold.
--again, this hinges on believing to know something is an act, rather than that such information is somewhere in your mind/brain.
Type 3 - pure reflective failure
I know my passwords to login to various online systems. But I don't necessarily know that I know my linkedin password. I may in fact know it but I would need to reflect on this to know whether I do in fact know it.
I'm sure there's several other types but it all hinges on how we define knowledge. If knowledge = the brain contains this fact, then knowing the fact is identical to knowing that you know the fact. If knowledge is an action, then until you do the act of knowing, you don't know that you know that you know. This may be difficult to capture because it's often trivial to do so when prompted.