I used to fly Europe to NYC on Icelandair several times a year a few years back.

Icelandair, I think, is partnered with Alaska Airlines. BA and other airlines fly there but are expensive.

Although my info is out of date, it used to be cheaper to fly US to Europe (or vice versa) with a stopover in Iceland, rather than fly directly. So take your girlfriend to London or Paris or somewhere for a weekend and stopover in Iceland if possible...