The wax candle has only two parts, the wax part and the LED and battery pack with switch. It took about 10 seconds to just pop the bottom part out of the candle. I’d probably go with the 4.5V DC wall converter as I know I can pick one of those up at the second hand stores easily and for less than a dollar. Cut the end off the wire coming from the converter. I would drill a small hole through the side of the candle, insert the wire through the hole and solder it permanently to the leads coming out of the battery box, and then pop the bottom back in place. If the light does not light you have the + and - leads reversed. The switch on the bottom will still work. You can see a picture of the innards here: http://tinypic.com/m/k2lsw4/1 Of course, never put standard batteries in the light when it is run off a converter.
I tested the light and it would not work on the Li-ion battery. I have done it often and this is the first failure. I guess it seems simple to me as I have dabbled with a lot of LED Christmas lights and LED bike lights where I didn’t want to use throw away batteries so replaced them with Li-ion batteries. Both the batteries and holders are cheap.
Sometimes the result isn’t very elegant but it works. A $2.50, 100 lumen, chip-on-board headlamp converted to a bike light. Went from 3 AAA to one 18650 battery. http://tinypic.com/m/k2lt15/1 and http://tinypic.com/m/k2lt1e/1 Even the cheap Chinese 18650 battery will run the light for about 2 hours between charges or switching out the battery for a fresh one.