We didn't find a weapon besides Sieghart's sword around there, so I'd imagine that either the killer used their own, and took it away from the scene... or maybe by the time they got there, the old man had already lost enough blood that the killer was able to take his weapon and fight him with it.

... But the only blood in the classroom was in drips, not splatters, so it didn't look like anybody got sliced up in that confrontation...

Maybe one person was using the sword and the other was just flinging furniture around? The desks were an absolute mess.