Neither of those wires can touch the handlebars or the control casing...otherwise starter will be engaged all the time..I think you need to undo the mounting screw and re-solder the green/red off the control (i.e. on the bench ), first brighten the piece and 'tin' it with a good hot iron ( let solder flow on it ) do the same to the wire ( tin ) separately , then hold them together and apply the iron until the solder on each piece melts together...hold very still until solder cools...do the same with the other wire, then maybe coat the connections with liquid electrical tape to insulate and ensure they cannot come in contact with anything when installed back in the casing and on the handlebars. Good luck, you can do it !
P.S....before you start, get the iron hot and 'tin' the tip with solder ( very imp. ) not a 'blob', give the tip a quick wipe to remove excess , must be just a coat of solder...then get to work heating the piece until solder will 'run' on it.....I think you got it !!