I assume you're familiar with J.R. Lucas's
Minds, Machines, and Gödel in which he argues exactly what you're pondering: is there something hidden within human intelligence that can not be replicated within the machine mind since the machine mind relies on formal, consistent, and axiomatic systems (where Gödel's incompleteness theory gets invoked)?
If Gödel's theories apply to machine minds but not to human minds for some reason then it must be impossible to recreate human intelligence in a pure machine.
If not, I suggest checking it out though I don't know if it is still in print. You can find Lucas updating his argument from the '60s in
this presentation at the 1990 Turing Conference. He makes many of the same points (in a more scholastically rigid form) as Dreyfus.
The underlying assumption of Dreyfus in
Gödel Escher Bach and Lucas is that the human mind is self-apparently capable of thinking any thought which is, to a great extent, unprovable. That since a mechanical mind can not be constructed in such as way to not run be constricted by Gödel's incompleteness theorom and we know that that human mind is not similarly bounded, then there is something unique about human intelligence that can not be recreated in mechanical intelligence.
Another issue when thinking about this is if you rely on too much on Dreyfus you'll be stuck in the 20-40 year old thinking on the issue. In recent years (particularly over the last 15), Dreyfus and Lucas's ideas have fallen somewhat out of favor as the Turing model for building artificial intelligence has fallen by the wayside and genetic algorithms and new approaches have been developed (though the Lucas school of thinking is confident they'll still run into a wall).
Of course they may still be right, but if you're looking into the ideas make sure you're reading more recent stuff than
Gödel Escher Bach.
I make no claim to know the answer or who is right when I read the debates. I'm just happy if I actually understand the questions they're asking. Personally, I'm pretty comfortable thinking that the human mind does not run into Gödelian limitations if only becaues it doesn't seem to work within a compete consistent axiomatic set which means the incompleteness theorem simply doesn't apply.
I have no idea whether it is technologically possible to build computer logic that also sidesteps the issue though the traditional way of thinking about computer logic as simply binary certainly does seem to run into Gödelian issues.