There's a wealth of stuff in your post that's extremely interesting, but I think we'll pursue this by email (I'm especially interested in the BOI files on the German activities to see how they compare to the German version of events). Drop me line with you email address so we can discuss this some more.

On the wireless issue - this is after all the topic of the post - I hear what you're saying. Two things: One, the Nauen specs are theoretical values. The ability of VLF signals to penetrate water decreases rapidly with distance. In addition to that, the German U-boats did not have towed arrays that would have allowed them to receive signals while submerged. Not that they submerged a lot anyway, which is why they had their aerials on folding masts. The Cap Verdes are well within the 3000 nm range.

Now, German comms with Africa (about 4500 nm away) were conducted by exploiting the "dawn phenomenon". This is, I believe, fairly unique to the HF spectrum and it works by bouncing the electromagnetic waves off the ionosphere during dawn/dusk time. it works better in a N-S direction - obviously, because the ionoshphere is is similarly charged along the meridians, but it works somewhat less well in E-W direction. The advantage is that you get by with a much less sophisticated receiver & aerial. Might be that agents tried this, but again, I have no records showing that U-boats were involved in this. It also doesn't explain why the German embassy made use of the State Department's telegraphic bureau.

The normal way of communicating "wirelessly" amongst warships or between the shore and warships in WW1 would have involved "relay stations" ie other ships if there was any great distance between the transmitting and the receiving stations. If you read through some of the radio logs you'll be surprised to see how crappy wireless comms where in WW1 even when ships were only a few hundred miles out at sea. Many U-boat signals were relayed by other boats. I can imagine that Mr Daniels' "conversation" (done in morse code) would have been relayed through a British station or another US warship, but that's me speculating.