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The light bulb box was a great idea, but I didn't account for his long tail feather, and it was a bit of a disaster.
I tried to close the lid down, and he came out the other end. Thankfully he didn't fall on to the floor :-(
I raided the kitchen cupboards and managed to find a trifle box, ironically, 'Birds' trifle :-/
I wrapped him up in tissue and put him in there.
I felt really sad seeing his little face. His eyes were still open and so was his little beak.
There appeared to be a seed or something in his mouth, so now I'm wondering how he died.
There was also a white/greyish liquid on the bottom of his cage, near to wear his body was, so I'm not sure if he was sick or something. Can budgies even be sick?
Anyway, it's done now. Thanks to all for your help and kind words.
R.I.P Reggie.