Now, now, let's not get contorted. If I recall correctly, the New Miyako Hotel is right on the South exit of the station. If you get into a taxi on the South exit, and he thinks you are going to the New Miyako, the driver will tell you to walk there. If you take a taxi on the Karasuma side, and the driver thinks you said "New Miyako", he'd tell you get off and walk to the other side of the station.

So, I think the lesson here, is get out the Karasuma side. If the driver looks intensely puzzled if you say "Miyako Hotel, please", just as much as if you said "please take me to the Westin" (the what?), then tell them you are going to Sanjo Keage. Better yet, tell them you are going to Sanjo Keage to begin with!