Although I am relatively new k-bike owner I can say for certainty, that the problem you are having may not be caused my the relay or the battery.
It has happened twice to me, each time my clutch had something to do with it! The first time it happened after adjusting the kickstand return mechanism (i must have did it wrong). The second time I got home from doing errands, turned the bike off and the starter continued to run. I figure it's the relay, so i go buy a 70 dollar non-returnable relay from Napa, they have the alternative as laid out in the Liebrary. Put in the new relay, and the starter continued to run... Now I have a "spare". Since the new relay did nothing, i look at the clutch - my locknut had backed all the way out (I remember when i snugged it up i didn't give it as much as I normally do).
However, re adjusting the clutch did not fix my problem, neither did changing the relay. My battery showed "low" on my auto charger (but, of course it did it was running my starter for a combined minute and half as I was trying to get test it and shut it off).
So, i charged the battery, removed and inspected the starter, cleaned the ground contacts connected to the starter, and the problem was fixed. Don't know what or why. However, i still believe the clutch had something to do with it. My battery tests good and my bike starts good every time, i think it was low AFTER the starter got stuck. My k bike is very clean and all my contacts were clean to begin with and I also do not believe my battery had anything to do with it.
don't go replace the starter relay before inspecting the clutch, and cleaning the ground points around the starter (and removing the starter if you have the time). Perhaps a low battery had something do with it AFTER i adjust the clutch, because by the time the battery had BECOME low.