Bend the coathanger into the shape you want
Solder LEDs, resistors etc onto a bit of veroboard. I used a female PCB header for plugging in the battery so that I didn't have to route long cables around the frame. Leave some length on the LED pins so that they stand up a bit from the veroboard
Fix your battery holder to the coathanger. I used sticky tape so I can easily remove it when the batteries go flat
Heat up some polymorph and use it to encase the LEDs (acting as a diffuser) and also fix the LED veroboard to the coathanger... you are done!