I translated this as "I don't know either František or Matěj" and, fortunately, that was accepted.
But could someone explain, please, why there is only one "ani" in this sentence, if the main translation uses "neither/nor"? I thought that would be expressed by "ani/ani." Thanks!
It is not accepted because it is simply incorrect in the sentence given.
If you use "neither" you need two "matching" words or clauses. In the given sentence, the "matching" bits are František and Matěj, two gentlemen whom we have/will come to know and love.
In your sentence, there is no match. It indicates that you "neither KNOW František nor Matěj" nor... what? There is no other verb to match up with "know." Placing "neither" where it is in your sentence works only if the sentence contains additional information, for example, "I neither KNOW František nor Matěj, nor WANT to know them."
I disagree. This seems like it is possibly a very british belief. This would be very common in English. It is implied... so 'I neither know Frantisek nor Matej' is understood as 'I neither know Franktisek nor (do I know) Matej'. This would be very common and natural. It should be accepted as a translation to English.
Despite your well-reasoned argument, to my native AmE ear, the sentence "I neither know František nor Matěj" sounds completely wrong, though I would bet it is very often used in everyday speech, especially in the US. I would not recommend adding it at this point, but times change...