Tegan, as a '71 650 Triumph Tiger myself who has had to go through my own personal apprenticeship of owning, maintaining & riding and old Triumph Meriden twin, here are a few things I would be looking at first.
Ignition: Are you confident the points and timing are correct?
Voltage: Have you measured it? These old twins demand a full 12 volts to get them started. Out of curiosity, have you tested horn or lights work just to ensure you positive earth battery polarity is correct?
If you have correct voltage, are you getting spark when you hold one of your HT leads to earth when kicking it over?
Fuel: How old do you think the fuel might be in your tank? If it is over 3 months old, toss it into your lawn mower & start with fresh stuff. Have you opened up the bottom drain on your Amal carb bowl drains to ensure that the fuel isn't contaminated?
Flooding the carbs: Probably the most important single issue when starting these old bikes. If everything mentioned above checks out OK, do you tickle the plungers on each of your Amals sufficiently to have fuel run down the side of your carb? Most new owners of these old twins are petrified of flooding their Triumph when the first get them. If your Amal float settings are correct, it is virtually impossible to flood them. If you think you have ticked the carb sufficiently, try giving the engine a couple of kicks with the ignition still off then tickle the buggers a bit more then turn on your ignition & see if it fires up when you kick it over. Everyones old Meriden Triumph starting procedure may be a little different but one thing my Tiger does not like is for me to twist the throttle prior to it starting. I have to flood it by tickling the plunger, and as I kick it over and hear the engine beginning to come to life, I then catch it with the throttle and give it a good rev into life. Twist it too early and it just wont start.
Good luck and I hope you get it started. Ain't nothin more frustrating than a kick starting bike that doesn't want to start.