Does this really have 4 switch settings for different intensities as indicated in the listing your posted? If so, it might have some kind of dimming ballast that has discrete rather than continuous adjustment.

Without pictures or diagram it is difficult to tell what is going on, but the parts that run the lamp (bulb) might be proprietary and unique to this lamp. If you cannot find parts from the manufacturer, it might be easier to modify the lamp than to try to put it back in shape. Where is the cover that you took off, in the base or in the top? Can you get the number off the lamp (bulb)? A 4-pin is probably pretty modern.

If you stick with the same lamp and socket, it might be relatively easy to gut it and install a new, single intensity ballast. It might be possible to install a dimming ballast. Either way could also might require some fabrication planning and skills with some sense of how to make it electrically safe. Going to LED is certainly possible as well with the same warnings.

Is it a Hammacher Schlemmer "The Eyestrain Reducing Floor Lamp" or VeriLux?

It definitely is possible to gut and re-wire, though you'd need electrical skills.

Thanks to both of you. It is the Hammacher Schlemmer lamp. No, I haven't experienced 4 settings, only on or off -- mine is a couple of years old. The parts are all in the top, and the cover that I mentioned is a curved metal piece that sits between the bulb and what I assume is the ballast.

Here is a pic of the inner parts, wiith that cover removed and base that the bulb plugs into also loose from its perch. I will put 2 more photos in follow-up; I don't have time to mess with photobucket right now. I apologize that it is a bit out of focus, this was my best of 4 attempts!

I will appreciate any instruction you can give. To me the only advantage to this lamp is the very bright bulb/high color temp. I otherwise prefer LED but haven't seen any this bright yet.

The ballast? no markings at all on the white plastic cover of this

This is the bulb

Thanks for your interest!

Here is a link that might be useful: bulb in question

You will just have to see if you can find a compact fluorescent ballast that fits into the space. Google will be your friend. Often these ballasts will power more than one type of lamp. Just look at dimensions, lamp, and mounting method and see what fits.