Every '71-'78 Triumph (and '71-'73 BSA) already has a "starter switch" - the button with a White and a White/Red wires. If your bike has the standard right-hand switch cluster with the kill switch, one of the black buttons has those wires; your bike's main harness does not have the continuation White/Red wire, you would need to run that alongside the main harness to wherever the starter relay is sited.
'79-on twins with a blank square bottom left of the right-hand switch cluster need the similar cluster with green button instead of the "square", Sparx sells them; Then aiui the main harness does have the continuation White/Red wire? If so, the other end of that wire will give a clue where Meriden intended to site the relay.
I can certainly tell you what is required electrically. The only thing I cannot tell you is how to fit the necessary larger battery - all non-electric-start British bikes with 12V electrics used the same Lucas 9 Ah (Amp-hour) battery, a Motobatt of the same physical size is now 11 Ah but Triumph fitted electric-start twins with a larger 14.5 Ah battery.
Standard '82/'83 electric-start twin timing cover? If so, there are several iterations of parts. The original timing gears were not designed for the torque of an electric motor trying to turn two 375 cc cylinders, they broke, Meriden strengthened them and something else broke, Meriden strengthened that, something else broke ... Meriden strengthened every component on the timing side except the crankcases ... then they went broke ...
Afaik, the twin starter was designed at Meriden in the early 1980's; Harris was/is a pattern parts maker; he did make a kickstart-only version of the twin in the mid- and late-1980's, under licence from John Bloor, who bought the rights to Triumph in the 1984 Co-op liquidation sale.
