First, let me reiterate what Turnkey said: Be very careful! The power supply can kill you even if it's unplugged, because the capacitors will hold a charge for several hours.
You can't really test the voltage for the tube as it's unlikely you have a meter that could test 20+kV. You can test the current, but that would be no different than what built in ammeter is showing (assuming it's working and the wiring is good.)
Honestly, I'd recommend waiting for the seller to get back to you.
However, if you want to do some troubleshooting right now, I'd say start by checking the wiring, connections and switches. Start by unplugging the machine from mains and then disconnect the three green terminal connectors from the PSU. Then, remove the sheathing from the wires so you can get a better look. The connector on the left is for mains power and connects to the red toggle switch, the one on the far right is the low voltage supply for the logic board and Low Fire input, and the center one has the Intensity, High Fire and the safety interlock(s).
It sounds like you are getting power to the logic board, so you probably want to focus on just the center connector and all the switches connected to it. You will want to test continuity for the switches and also impedence on the potentiometer.
Given the wide variety of how the K40s are setup, it's hard to be more specific. For instance, I don't know if you have a water flow sensor or if there are interlocks on your build area cover.
Again, please be very careful and stay away from the thick red wire from the PSU to the tube and the PSU in general. Unless you really know what you are doing, there is a very high risk of injury or DEATH messing with the PSU and you really wouldn't be able to fix any fault.
regards,
aj