The 2010 Roadster initially had the same electrical config as the Earlier (pre-Roadster) models which has the Headlight Current through the Key-Switch
Yours has obviously suffered burned terminals on the connector and was bypassed.
Thing is, the 2010 had a recall to install a retrofit relay that bypasses the current to the headlights OUT of the key-switch circuit.
Yours may have not had this done and you initially had the connector failure and now have the key-switch failure.
You should not need EB bypass kit and instead have the recall which should be done by Triumph at no charge (if not already)
Regardless, the EB kit will not FIX the fault, only stop it from happening again if you replace the switch.
And the EB kit is superflous if the recall has already been (or will be) done (but sounds like it probably has not)
The switch is actually nicely accessible once you prop the tank up. But it's riveted in. ....
It's not actually riveted - they are bolts however they are sheer bolts, where the heads snap off on assembly
So they will actually screw out if you can make some slots in them (otherwise you would drill the rest of the head off and then replace with a regular screw)
EDIT - Whoops! Scratch most of that - don't know why I misread & thought was Roadster
The switch may have failed, especially if non-std bulb has been used in the headlight (or additional lights wired into the OEM circuit)
But much rarer to fail on a Touring (although HAVE seen that connector fail)
I have a post on here for a very simple bypass for the headlight with a single relay.
I can find that for you ...........