The Waldorf Microwave XTk is the keyboard version of the Waldorf Microwave XT. It uses the same line-lump power supply, only it’s hidden inside the unit. The coaxial power connector normally used on the XT is covered up, and the power supply wires are soldered directly to the board.
There are many things that can make these synths not power up, but one common culprit is the transient voltage suppressor diode that is connected across the power input. This can fail to a short, causing it to appear that the power supply has died.
To troubleshoot, take the unit apart and disconnect at least one side of the power supply. Plug the power supply in. Does the light come on now? Do you measure ~12VDC at the output? If not, then the problem is probably a bad power supply after all. If so, then the problem is somewhere else in the synth.
Next, lift one leg of the transient voltage suppressor diode and use a meter to take an ohms reading across it. It should either appear open or have a very high resistance. If the resistance is low (like less than 1 ohm), then the part is bad. This very low resistance shorts out the power supply and makes it shut down.
Obtaining and replacing the diode is fairly easy and inexpensive. The part number is BZW04-15B and Mouser stocks them. Don’t worry about installing it the wrong way; it’s bidirectional.