I believe that Mbed doesn’t have any working built-in support for using a J-Link. You will have to use the J-Link command line tools to flash the bin or elf file that got built. Maybe the ST-Link isn’t showing up because there are drivers or Python dependencies missing?
For what it’s worth, mbed-cmake does have built-in support for flashing Mbed projects using a J-Link, and it also works better with flashing custom boards.