The Gene Krupa Sound King would be in a 5x14 or 6 1/2 x 14 configuration, would feature a brass shell, with 8 or 10 lugs and a Zoomatic (tombstone) strainer. In 1965, the company reduced the size of the knob on the Zoomatic strainer and stamped a large "Slingerland" on it.
Around 1967 they introduced a "Dual Super Snare Strainer" but this strainer was no longer catalogued after 1976. I doubt that the drum you're looking at would have that particular strainer.
If original, the drum should also have triple-flanged chrome over brass hoops which flange inward toward the middle of the drum. These are known as "stick-saver" hoops. There should also be a dial-tone muffler assembly mounted on the shell on the panel to the left of the badge.
I'd also take a closer look at the badge to see if it is, in fact, a black over silver in lieu of a black over brass. My assumption is that it's a Niles, Illinois badge. See if it has a serial number on it.
Good luck!Keep on Pl