Below is the start & charge circuits as ripped directly from the Triumph workshop manual. We know that the starter motor's working and the solenoid's not clicking. So following the circuit diagram we can work backwards to find the fault.
Firstly there are four fuses to check, numbers 11 (30A), 2 (30A), 5 (15A), and 9 (15A). Don't rely on visual inspection, test them with a resistance meter.
Check that all electrical connections (including the battery terminals) are tight and are not corroded, dirty, or wet.
Measuring voltage at the solenoid you should find 12V on the large gauge brown wire and 0V on the Black & White one. With the ignition on, gearbox in neutral, you should find 0 volts at solenoid terminal 1 (black wire) and this should change to 12V when you press the start button. If so the solenoid's at fault. If not...
Test alarm connection 3 with the start button and ignition on. You should find 12V there. If not...
Remove the starter relay. Test between pins 6 & 4, you should find 0 resistance. Using an ohmmeter (multimeter set to resistance) test between ping 1 & 8 and you should find and open circuit (infinite resistance). Get a 12V battery (car battery will work fine if you don't want to pull the bike's battery). Connect the relay's pin 4 to the negative terminal and pin 6 to the positive. You should hear the relay click. Measure resistance between pins 1 and 8. If you don't find 0 resistance the relay's not working.
Check voltage on either side of the start button & kill switch...
Check voltage at ignition switch. You should find 12V at pins 4 & 5 when off, and at all four pins when on.
Finally, to test resistance on the diode pack you should measure 0 resistance from pin 2 to pin 1 and pin 2 to pin 3, and if you reverse the connections you should measure an open circuit (infinite resistance) from pin 3 to pin 2 and pin 1 to pin 2. If any of these aren't true replace the diode pack. Test resistance between each of the switches and earth individually.
That's the entire circuit. If it's not any of those things your machine is possessed by an evil spirit and you need a priest.
Good luck.
