Soldering problem with heavier components
Short answer: you need a better soldering iron.

Longer answer:

Your little battery-powered soldering iron doesn't have enough power to heat a large mass of heat-conducting metal up to the melting point of solder.  All of the heating energy from your tiny iron is being conducted into the mass of metal that you are attempting to solder and being radiated away.

I would suggest that you need at least a 40 Watt iron to solder 1mm thick wire of any significant length.

Others might suggest that you would need more power than that but it all comes down to what soldering iron you use.  A really inexpensive iron may not work well just because the design is such that there is significant thermal resistance between the heating element and the object being soldered.  But better irons will do the job nicely.

A specific example of a soldering iron that works well at 40 Watts is the Metcal MX-500 series with a sttc-137 tip.  This iron will solder the entire perimeter edge of a male DB-25 connector to a piece of copper-clad PCB material using 63/37 solder.  It takes a while but it will get the job done.  Your typical Radio-Shack 40W iron won't even attempt to work.