I would try this
1. (I presume that blinking mbed program is still running on your board). Reset the board and observe the voltage at the red led output pio0_7. If it has 50 hz square wave the program is running and the clock and the MCU are alive. (if you don't have an osciloscope, measure the dc voltage, it should be half of 3.3v).
2. Measure the voltage on the MCU PIO0_1 and see that the voltage changes between 0 and 3.3v when you press on the ISP button. This is required to enable the ISP mode.
3. Verify that PIO0_3 input of the MCU has 3.3V. This is required to enable the USB ISP mode.
4. Use a chopped USB cable to test the usb data path from the cable to the MCU (via the 33Ohm resistors). Make sure there are no shorts.
The connector on your board felt notchy when I inserted the cable. I assumed it got extra flux or solder paste, but it did work.
Let see if this will help.
If anybody is interested, schematic here.
https://github.com/zapta/arm/blob/master/pro-mini/eagle/arm-pro-mini-schematic.pdf?raw=true