Purple Haze,
Sorry a bit late picking up on this thread. Some mind numbing statistics been given though.
Going back to your question. The procedure is really quite simple.
You would gradually descend to an altitude at which the aircraft could maintain altitude on its one remaining engine. ( A 3 or 4 eng' a/c would do much the same). This process takes a lot longer than most people think.
During the above you would start the process of diverting to a suitable aiport. The decision of airport is subject to continuous assesment during any Etops flight. The airport chosen (in your Scenario) would probably be around 2 hours away maximum at the single engine flying speed.
The remaining engine will probably work fine for years notwithstanding the couple of hours you require. The fact that the first one failed is for most Pilots a once in a lifetime event.( Most, not all!) If it really isn't your day and the other engine fails, then your choices are really made for you.
It is interesting that there are more critical scenarios from a fuel point of view. For example a depressurization ( regardless of how many engines you have) will in all cases require a descent to around 10,000 ft. At this altitude those engines burn a lot of fuel. The weather is often quiet unpleasant as well. There is a certain irony that if the engine also fails in this scenario you would be better off from a fuel standpoint than if they both kept running.
As others have said the rules concerning twin engine operations over extended distances, limit the allowable distance from a suitable diversionary airport. These rules also take into account terrain factors where they may be applicable.
There is a lot of opinion on the rules concerning ETOPS however the point I am trying to make is that there are far more difficult scenarios to contend with than a twin engine aircraft having a single engine failure in flight wherever in the world it happens to be.