I used these bulbs:
I now have half a dozen in different fixtures and have been very happy so far - good price ($2.50 ea. in the six pack), nice light output, and good build quality. They can also be modified to put the pins out the back instead of the side, though you may need longer pins.
The transformer I pulled out of a 12V 1A wall wort, it had fallen out of the wall one too many times and the secondary leads cracked out of the rectifier board. Since I don't need DC, soldered on some new leads and tossed the rest in the junk box. Clever zip-tying and foam weatherstrip application make a firm mount that should dampen any vibration the transformer develops over time.
The lamp still has the ballast for the fluorescent tube in it, but I bypassed it out of the circuit. Not sure how that would play with the power transformer.
Then it's just wired hot to switch, switch to xfmr, and neutrals together. Secondary wires are just daisy chained to the LED boards in parallel.
I used some 1/2" foam weather strip to space the boards off the lamp (there's an electrolytic filter cap back there to get in the way) and then some 3M double-sided foam adhesive tape I found in my sticky-stuff drawer, and there they are.
Nice project! I've got two of these old lamps... one of the newer style like yours, and an old battle worn desktop version that has to be from the 60's or 70's. Both of them could use an update to LEDs. Ditching the EMI that florescent lights give off would be a nice added bonus!