It might actually be a 1 1/8″ headset, as I think Kona were one of the first companies to move to the new standard. regardless, a ‘standard’ threaded headset will work on it fine. Have a look for older Shimano threaded headsets, as they were very good quality.
If the cup is moving in the headtube, then it could well be that the headtube has flared outwards. This could possibly be remedied by bunging the top of the tube in a vice or something and GENTLY squeezing it back into shape (steel frame so not a big deal), but it would be a bit tricky and has the potential to go horribly wrong.
I have successfully used Araldite Rapid Steel to ‘glue’ a headset cup into a frame. A bit too successfully, as the frame is now dead and I can’t get the headset out…. 😳
Loose lower cup might be symptomatic of repeated heavy landings on the front end. Best to check the steerer tube where it meets the rest of the fork, for any signs of damage.