Well...turns out my firmware is up-to-date (v.2.19), so I don't need to update. I thought about downloading and flashing just to experiment, but it ain't broke, so...
I've tried two wireless keyboards now with the same result - qwerty part doesn't work. The only real use for the kb is when using the Opera web browser in Smart Central mode. Using the remote to navigate the virtual keyboard and type a URL is excruciating! A wireless mouse is the next best alternative since you can bring up the virtual keyboard (press menu on the remote while the cursor is in the web address bar), and then move the mouse around and click on letters one at a time; it goes pretty quickly with the mouse. The Pandora and YouTube apps have their own interfaces for character input, so the kb is of no use in those apps.
The first kb I tried was a Rii i8 mini wireless, which resembles a game controller, but with a full keyboard, mouse touchpad, and directional arrow buttons. It uses a USB dongle. Everything on it works except the kb. The second kb I tried was my wireless kb and mouse from work which also uses a USB dongle. Again, everything worked except the kb itself.
I have now emailed Sharp support asking for examples of wireless kb's that DO work. We'll see if they reply.