How do I hack my soldering iron?
Rather than do all that engineering and risk damaging the soldering iron, for such a small load (120W) I'd suggest just getting a commercially available 120V to 240V transformer or adapter in the 200W capacity range.  They're not expensive. 

Also, there's a provision for installing 220/240V outlets in North America. It uses these outlets made for 240V which install exactly the same as common 120V outlets, except keyed different. It's called NEMA 6-15.  240V breaker and you're done.