Sadly you were missold on the device and I suspect it is past the return period. Fortunately it is possible to rectify it with modern technology.
You will want a floppy drive to USB adapter. I find them in old boxes of tech gear all the time but new are also not so bad.
A raspberry pi with 3.5mm audio out, including whatever else is needed to get one of these working. Can substitute for a PC/laptop if you have one of those around, though presumably if you can afford the number of D cells needed to power the beast in the opening picture then you can afford a raspberry pi. There should be many cases of media player software available for such devices. Getting it to autoplay is a bit harder but still within reason, and if you want to kick control to a phone then all is good.
A 3.5mm to FM radio rebroadcaster. Personally I do like the Chinese offerings here as they do tend to be less concerned with acting as a legal device (strictly compliant versions are not always the best) and instead working. At the same time if you use the USB charger on them they have a habit of bursting into flames so be aware of that one. With this you can tune the FM radio element to the rebroadcaster.
Alternatively you could solder a bluetooth receiver inside the radio and broadcast over bluetooth instead.
Be aware that on some older devices you will have to enable the audio out if you tap into the CD player aspect here as some only play when playing a CD. Some will cheat and burn a CD full of blank tracks.
EOF so song