Out of curiosity, labeling aside, is the ballast actually in the wall wart power supply or in the lamp base? I ask because that lamp has an awfully substantial lamp base, suggesting the bulk of the ballast could be in the lamp base and the wall wart may be little more than a power supply (sometimes the parts are labelled as a package rather than as individual parts). I'm not terribly familiar with the Ott brand lights, but at their core they are a normal fluorescent bulb (with phosphors optimized for a certain spectrum) and I would expect them to use an otherwise typical ballast.
It is often possible to repair a damaged ballast -- often times a component (fuse, resistor, diode, etc.) burns out when the bulb fails -- depending upon exactly which component failed. Many are practical to replace but if by chance the inductors fail then it may be impractical to fix. The real challenge can be gaining access to the ballast as most of the wall warts nowadays are ultrasonically welded plastic which can only be accessed by physically cutting the unit open. The challenge then is safely resecuring the power supply housing after it has been repaired. If the ballast is in the lamp base, which appears to be secured with screws, then access for repair is much simpler. Whether or not repair is practical and worth attempting depends largely upon the overall replacement costs -- or novelty value -- since they often fail at the bulb's end-of-life. The other challenge -- in the absence of a working unit to reference against -- can be trying to identify the characteristics (e.g. resistance) of the failed component if its markings have been destroyed by the failure -- it is hard to replace a component whose correct value you can not quantify and the manufacturers seldom provide schematics for their products nowadays!
Sadly, we have become a senselessly throw-away society and it absolutely drives me nuts!
One option would be to scan sites like ebay where both working and non-working (e.g. 'parts only') equipment can be bought and sold. A quick and dirty way to get your unit back up and running for a reasonable price is to buy someone else's junk. Otherwise, I really don't know where to look for identical drop-in replacement parts for your Ott light since no vendor specifications are available to base such a suggestion upon. The typical advice in your scenerio is to buy replacement parts from the original manufactutrer (OEM), but if they will not sell the individual parts then you are left with either replacing the unit or attempting third party repair.
Best of luck!
