Returning to the thread - I believe I've sussed the problem (mine anyway) principally from study of Place of articulation - Wikipedia, the free encyclopedia and
Apparently, I form my 'k' with an epiglottal/pharyngeal articulation (locations 10 & 12) using my neck muscles rather than my tongue. Not only does this articulation lack the necessary tongue movement for multiple tonguing, but repetitive constriction of the throat leads quickly to tension in the neck and shoulders and a squeezed airflow. Since I form hard 'g' in a similar location, maybe even further back, this also explains why I never had any joy with "da-ga-da-ga-da-ga" either.
After a bit of practice, I find I can raise the back of my tongue to give a contact roughly between locations 8 and 14, relax my neck and get a series of reasonable velar plosive "ku-ku-ku"s down the instrument.
This 'arching', though a bit strange at first, doesn't cause undue strain. It also puts my tongue tip naturally at the gumline of my lower teeth rather than above them, which seems to ring a bell with some other bits and pieces I've read recently.
It's a pity I didn't find this out 40 years ago!
Now I've to decide whether it's worth investing a couple of years' practise in developing a technique I've managed to do without until now.
Many thanks to all contributors on this thread. It's been a great learning experience for me and I am genuinely most grateful.