This was a Kitchenaid KCM1204 filter-coffee machine, I bought it off Ebay cheap because its display had dimmed to the point of being unusable.
Turned out that the low-voltage power supply was supplying too little current, because the "dropper" capacitor (connected directly to mains live) had degraded. This is a known problem - e.g. from this post on EEVblog :
"Class-X capacitors are deliberately made with thin metallization so that they will self-heal easily. This limits their series current capability and causes them to lose capacitance over time. This is a common failure mode in capacitor dropper PSU applications."
The capacitor from mine is marked as 0.82uF, but now measures only 0.36uF. Time to replace it...

To repair:
1. *** UNPLUG THE MACHINE FROM MAINS BEFORE STARTING WORK! ***
2. Take the bottom panel off the machine (6 Philips screws).
3. Trace the mains cable, it goes into a small plastic enclosure - undo 2 screws holding the enclosure into the machine, pop the two halves of the enclosure apart.
4. Identify the dropper capacitor - large rectangular component, marked "X2", "280VAC", "0.82uF" in the case of this machine.
5. Purchase a replacement part of the same voltage-rating, capacitance and pin-spacing - ~£3 / $5 for one-off.
6. Desolder the original capacitor and solder in the replacement. Reassemble and refit the power-supply enclosure, then refit the bottom panel of the machine.
7. Plug the machine in - the display should now be readable again.

 

The same problem is the common failure mode for socket mains timers - the timer keeps running on the backup battery, but the relay power has gone. I've repaired many by replacing the dropping capacitor.

 

Full disclosure:
A week later, I bought another machine of the exact same model (for even less money!) with the exact same symptom - dim display but otherwise fully-functional. Full of confidence, I replaced the power-supply capacitor - it made no difference! And then measuring the capacitance of the component I'd removed, it turned out to be within spec.
On this machine, it seems the backlight LEDs (three tiny 0603 types) have just dimmed over time. Not expensive to replace, but it was difficult to get the front panel off the machine (broke several plastic clips) and then needed some fine soldering under a microscope.