Here's a favorite of mine (although I admit not of Mr. Tb's who finds the rooms too small):
Hotel des Deux Iles
It's located on the Ile St. Louis, just behind the Cathedral de Notre Dame. The hotel is quite comfortable (despite the small rooms), the neighborhood is charming, and it is quite convenient for both the right and left banks.
I probably shouldn't mention this, as you're travelling solo, but it's also very romantic, walking back from the left bank at night, across the bridge, with the view of the Cathedral...
Mmm, sorry, didn't read your full post. This may not be luxurious enough for you, but I do agree that the Ile St. Louis is THE place, when you're on your own (or looking for a romantic setting).