Logitech K800 keyboard turns on and off continuously. help!

Hi, I have a Logitech K800 keyboard that I love and it's impossible to buy another one like it. Unfortunately I have to change it for only one reason: the backlight turns on and off continuously, even when the PC is turned off. So its battery lasts no more than 3 days. Do you think there is a way to fix it? I also tried replacing the battery, but the problem remained.Thanks for some advice.
 
Can you find a used one on ebay or similar? Not much you can do with keyboards but trash them and buy something else.
 
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