Non-breaking spaces are often added because they are present in word processing docs and then copied into the WP editor. I find the easiest way to remove them from a single post, including any unwanted formatting that results from copying text from a Word doc, is to copy the text (in visual editor mode) where they are contained into a text editor (thus stripping all formatting) and then copying back into the visual editor.
To remove them from all database content I would export the SQL database using a database management tool. I use phpMySQL to do this – select the table and click export button, set a compression format and also set it to include the command to drop tables on import so the import replaces the existing data with the new when reloading the DB. That way you get to extract the file from the downloaded zipped export so you have an archived backup to fall back on if something screws up.
Then, using a text editor, do a find on and replace with a space, and re-import the edited SQL database. It only takes a few minutes.