Hmm that’s odd. Well I wouldn’t just sell it. If you’re really stuck take it to a local shop. If you want to keep troubleshooting then it may help if you explain exactly what is happening when you turn the key on. Does the engine light come one, if so does it stay on or go out? If it stays on try checking the ECU for codes, though I doubt there is any. You say the relay does not click? I’m guessing the fuel pump will not prime for 2 seconds either? Based on what you have described so far it sounds like the main relay is not working and wiring is likely the problem.
You can test the wiring to the relay:
There should be 12v at the yellow/white wire at all times.
There should be 12v at the black/yellow wire with the ignition on.
The black wire should have continuity to ground at all times.
Let me know what you find.