1) Try connection the pi to a monitor or TV via an HDMI cable and see if you see anything when you power it on. If you see stuff when you connect to a monitor - the Piper screen or the supplied cable may be bad.
2) If you can, see if you can find someone with a PI. Ask them to test your SD card on thier PI and test your PI with their SD card.
a) their PI - their SD card = this is the control and should work
b) their PI - your SD card = the SD card has an issue
c) your PI - their SD card = probably issue with your PI
d) your pi - your SD card = probably issue with your PI
3) if things work with their PI, connect that pi to the Piper screen and see if it works.