Thanks for the suggestions and links.
The LCD display works fine, as does the contrast pot, so it's just the backlight. Tapping doesn't make it light up, so it's probably not a solder joint issue. Only once in a blue moon will it light up at power up, and it's completely unpredictable when that will happen, might be tomorrow, might be next month. So I suspect an intermittent component in the backlight circuit.
Getting a replacement for about 80 AUD probably is the best solution. But I am loath to dismantle it because I play it all the time. I don't use the accompaniment features since I learnt to play two-handed. So it's more important to me to have a working piano than to read a display that I don't use at the moment. I can shine a torch on it if I need to change anything.
So I'll take note of the various suggestions and maybe someday when I get a better piano I might fix this one. Thanks again all.