Firstly, welcom to Ribnet!
I suspect the reason it dies after 15 mins at idle is that it has oiled itself up. I can't think of any premix 2- strokes that wouldn't.
So, assuming you have a well tuned carb, it's sparking healthily and it just won't start, the list that comes to mind is:
- Deadman in place? (not an insult- we've all done it!)
- If you replaced wires, are the cyls sparking in the right order? (i.e swap the plug leads around)
- A blueband will have good old condenser / points ignition - is your condenser gubbed?
- Not sure what carb you have on there, but if it's one of the ones that doesn't have a choke flap but the choke works an "enrichener valve", I find mine will only start from cold with that open (i.e "choked") and the throttle closed (even tho' the throttle has a "start" position that would imply you want a half open throttle).
Try that lot & let us know what happens.