Look to what is driving the output tube - both the power supply feeding it, and the upstream components in the output section (voltage regulator, etc.).
Cost will be dependent on that. I would think $250 on the low end, and potentially 3 - 4 times that on the other end of the spectrum. I know that's a wide range, but I don't know how Audio Note components are constructed. Whether discrete components or an entire board(s) needs to be replaced determines a lot, as does the time/labor involved.
As Davidbigeye said, the unit should work fine through the digital out. And, as Rbrowne said, contacting the service rep is the right move.