As far as I can make out.

Air Defense Intelligence, set up by British Air Ministry.

To report on basically every thing to do with the air war.
And as ((K) seems to refer to only Luftwaffe, I would say that it was used to denote anything to do with the Luftwaffe.

A B C D etc would possibly refer to other reports on various systems of Air defense, such as our own radios radar etc.

Sample of A. D. I. (K) Report No. 393/1945

http://www.cdvandt.org/ADIK393all.pdf