The short segments and the decimals are all driven by IC U3. If none of them are lighting, then either U3 is not running or it is not getting the signals.
Check that the chip is oriented properly and fully seated in its socket. In the schematic ( https://bcdn.evilmadscientist.com/source/alphafive/fiveletter_RevC.pdf ) note that U3 has power and ground connections at pins 1 and 16, and receives its serial data with pins 2, 3, and 4. So, check that the soldering is good at each of those pins. Next, check pin 22 of U2, where the serial data comes from.
