That extra connector that goes nowhere is for the factory alarm system and/or testing. Don't worry about it.
At this point, you have no idea whether there was a problem that caused the previous owner to remove the signals, whether the correct flasher is installed or if you've miswired your installation of the LEDs and load resistors.
Nothing like making it simple for remote troubleshooting!
Turn signal flashers (relays) work by sensing the load applied by the turn signal lamps.
In the old style electro-mechanical flashers, the load current flowed through a heating element which caused a bimetallic strip to bend and make/break the contacts that applied power to the lamps. When the load current decreased, the flash rate changed depending upon the internal mechanical structure of the blinker.
The newest euro standard flashers are completely electronic and sense the voltage developed through an internal current sense resistor which tells the control chip how many lamps it's flashing. As a safety feature, the control chip reads a 'too low' current as a lamp failure and increases the flash rate to alert the operator of a problem. If there's a short in the flasher circuit, the chip reads a 'too high' current and will limit the current output so the wiring and blinker don't burn up. This sensing mode also requires that the flasher must be match to the load it's supposed to drive or the load sensing doesn't work properly.
Between the old mechanical blinkers and the newest electronic models there have been a bunch of hybrids and variations, so it's impossible to know exactly how your blinker is supposed to function in a given situation and to diagnose a problem from the information given.
The Triumph blinker (whatever the internal system) uses a three terminal blinker with a ground connection, a +12V connection and a pulsed output. In operation, the +12V power is switched on and off in the blinker and applied to the output terminal. When either the turn signal or the hazard switch is actuated, the output of the blinker is applied to the lamp circuit(s).
According to the schematic, the ground to the blinker is a Black wire with a Yellow stripe (Black/Yellowl) and the +12V is applied via an Orange/Green wire. The blinker output to the lamps is an Orange wire.
At the turn signal and hazard switches, the circuit breaks out to a Green wire for the left side signals and a Gray wire for the right side signals. The turn signal switch connects to one wire (Green or Gray) at a time to the Orange wire, while the hazard switch connects both wires simultaneously.
At the lamp connections to the wiring harness, the Black wire is the ground lead from the lamp and connects to a Black/Yellow wire while the White wire connects to either a Green or Gray wire depending on the side of the bike.
Since the LEDs are lighting when they're supposed to (but not *how* they're supposed to) they're probably connected correctly.
The load resistors (that simulate tungsten lamp loads) need to be checked. They should be installed parallel to the LEDs -- one side to the Green or Gray wire and the other end to the ground or Black/Yellow wires. Check that your connections are firm and that there's no exposed metal to short out against the chassis or another connection.
If the LEDs and resistors are correct then the problem may be in the left grip switches, the wiring connections in the headlight or a defective/incorrect blinker.
My highest probabilities are that you've got a miswire with the load resistors or that the blinker is either incorrect or cooked.
Check out your resistor installation and get back to us for further troubleshooting.