This shows the problem - the solder melts from the heat and the wire lets go.
If the switch is not repairable, instead of having to swap the entire switch and live with two keys, you can separate the mechanical & electrical halves and retain the original mechanical lock/key with the new electrical switch/harness.
(This pic is from a different model Triumph but the problem is the same)
That carries the total current to the headlight/starter circuit plus the Ignition circuit
The headlight circuit is ~ 10A and the Ignition circuit ~ 4-5A (mostly for the Fuel Pump, plus some other lesser loads)
to much. the ign switch usually gets hot because of the light circuit and melts off the soldier on the white wire and then you are walking. i did the measure amp thing but with 1 daymaker and one regulator. it is probably in this thread some where.