Hi Chris,
The tube won't pull the load, even with high impedance headphones. Zout of the GK-1M is around 120R, which is not adequate for headphones to about 1K impedance, and will incur sonic degradation with 'phones to about 2K.
You need a buffer for best results. There are many ways of doing this, but most involve a push pull Class AB or single ended Class A output stage, using semiconductors rather like the drivers in the AKSA power amps. You can also use an opamp, but for reasons of preference I'm loathe to suggest this approach, though it works just fine. Perhaps the best way would be a Class A push pull bipolar transistor output stage, using an opamp as a servo for offset control.
Given a bit of time, I can give you a proof of concept circuit, but there would be some work involved to debug it. Are you willing, Sir? Email me privately.......
Cheers,
Hugh