That clicking means that your not getting enough power to turn the starter. The batterie is probrably almost dead.
If it's that cold outside, then you need to warm up the batterie and recharge it.
New batteries arn't always fully charged, just enough to turn over the starter and get the alternator to start providing power.

It's -25 degrees below freezing tonight in Fairbanks Alaska.
Cold weather will suck the power from a batterie in hours if not kept within operating temperature.

I'm a security guard and I have to jump start vehicles at our corperation every day with the same sound as yours.

I also get that same sound every morning from a dead batterie if I do not plug in my vehicle. My Sporty is winterized with a batterie warmer, oil pan heater, transmission heater, and engine block heater.

You should be able to jump start your Sporty after allowing the batterie to suck the life out of the one your jumping it from. Just have the other guy hold his revolutions at about 2000RPM while you attempt to start you vehicle so you know his alternator is going and providing enough cold cranking amps to start yours.

Good Luck and keep us posted.