First of all, just connecting the SWD wires doesn't do the trick. The nucleo's SWD header cannot supply power to an external target, it will just sense it (if I'm reading the schematic correctly). So an extra 3.3V wire must be added from one of the 3.3V pins on the nucleo board to the breakout board's VDD pin.

I also had to unplug/plug the nucleo board's USB cable before it would let me program the chip with blinky code, but now it runs:

The LED is too bright. I used a 180 ohm resistor, so 330 ohm should be sufficient.