I know Rick Steves loves the Rue Cler neighborhood, but IMO it's rather out of the way for many of the sites that you want to see in Paris. My preferred location is in the 6th arrondisement, as close to the river as possible. We've stayed in that area several times and walked to most things. Second choice would be the 5th, followed by the 4th. Here's a list I put together a few months ago for a relative who was going to Paris for the first time:
Note on bed sizes: most will be doubles and will be noted as 140 cm. Queens are 160 cm, kings are 180 cm, both are uncommon
http://dauphine-st-germain.com/en/index.html - nice location, small rooms, well rated on TA and here https://www.oyster.com/paris/hotels/hotel-dauphine-saint-germain/
http://www.hoteldelesperance.fr/en/ - decent location but appears pretty basic
http://www.hotel-britannique.fr/en/ - not my preferred side of the river but looks charming, reviews are great, could possibly get in around $200/night
http://www.hotel-collegedefrance.com/en/ - good location, great reviews, could possibly get in around $200/night
http://hotel-diana-paris.com/en/ - good location and price, but the standard room is very small (about 110 sq ft)
http://www.hotel-grandes-ecoles.com/index-en.html - decent location, appears to have an outdoor space
http://www.hotelcaron.com/en/ - great location, true queen bed, many positive reviews, doesn't say what size the rooms are
http://www.grandhotelmalher.com/index-en.html - good location, modern showers, more modern rather than Parisian charm, room size not listed
http://www.mayet.com/ - more contemporary in style and location is further east and further from river but price is lower
http://www.paris-hotel-stjacques.com/en/ - good location, lots of parisian charm, rooms a little larger than some
http://www.hoteljardindecluny.com/en/page/hotel.2.html - great location, actual queens and kings available (avoid the classic room, it's very small), modern chain in appearance
Have a wonderful time!