To make absolutely sure I answer all your questions, let's go through.
"Anyone have additional specific personal experience/information on the Rembrandt?"
Given in my first post.
I do swear this is personal experience and my husband and I have stayed there recently.
"Size of rooms, compared to similar hotels? bathrooms?"
I have given you the size of our bedroom and bath. Furnished pictures.
I would post my very own pictures taken there, but we cannot do that on the Rick Steves forum/website.
Not to my knowledge, but do call or email the hotel with any questions you may have like this.
There may be something planned for the day you arrive there that we on this forum would have no knowledge of.
Yes, get a room in the back of the hotel if you want complete quiet.
"What sets it apart from its competition in its neighborhood?"
The hotel is very nice and clean, almost luxurious.*** Some in the neighborhood are not.
Rembrandt's price is very good for getting a clean nice room.
You can go up, up, up in price in this neighborhood for the same quality.
The staff is friendly, polite, helpful. At other hotels in the neighborhood (one in particular) this is not true. You can run into some pretty uncaring staff in some hotels; won't fix problems with the room or move you, are surly when you approach the front desk to ask questions or directions.
***Notice I put three stars by the word to qualify it. There are certainly luxurious hotels in London, with prices as high as the sky.
What many people on this forum look for is a very, very nice hotel with a reasonable price.
A really low price guarantees a dump. A really fancy and very large hotel room and bathroom, the size you have at home in the USA, guarantees a price so high per night it will make your eyes water.
Again, your question, "What sets it apart from its competition in its neighborhood?"
You can do some homework and research online yourself.
Rick Steve's London guidebook names a number of hotels in this neighborhood.
You can also Google search "London hotels near Victoria and Albert Museum".
This will give you some hotels to research yourself.
None of us have researched or stayed in every single hotel in the neighborhood, so we can't tell you about how every one of them stack up against Rembrandt.
You will enjoy doing this research yourself, as we all do before our trips.
If you want more suggestions for other hotels in this area, go to the England forum and post a new question. You will get more responses there.
(You have posted this question under "England Reviews" which is not the best place for your question.)
Hopefully, some more people who have stayed at the Rembrandt will chime in here and give you more information.