If it's been sitting unpowered for years, I'd first open both the rig and the power supply and clean them out. You might find insects, spider webs, dust - lots of things that can cause shorts and even fires. (Remember, this is tube equipment, so it's running on high voltages, not 12 volts.)
The second thing I'd do is check the electrolytics. Or, if you can afford it, just replace them all. If not, I wouldn't use a variac - you'll be running high voltage (not full, but high) on the plates and screens while the filaments will be underpowered - an easy way to destroy tubes. You could use a variac on the power supply alone - with a dummy load (a high-powered resistor) on the high DC voltages to keep things safe. But I wouldn't do that with the KWM2 itself connected. Unless you like fireworks in a box.