Roppongi is the most foreigner friendly, but mostly for the bar / restaurant scene. I wouldn't necessarily want to stay there, just take taxis in and out for dinner.

Shinjuku is a very convenient transport hub, as well as near some of the interesting sites (Meiji Shrine, Tokyo Metropolitan Building viewpoint, etc.). I quite like staying in that part of town.

Since you get 2 nights at the HR on points for the price of 1 at the GH or Andaz, why not stay 2 or 3 nights at the HR and 1 or 2 at the iconic Park Hyatt? It's a short walk or cab ride between the two.

P.S. Walking around Ginza, Shibuya or the east side of Shinjuku station at night when jetlagged is great for sensory overload. I feel a bit cheated when I'm in Tokyo but sleeping normally.