How can I mend my ceiling light fitting?

Sorry, I know I'm going to sound like an idiot but I can't help it - girls aren't taught this sort of thing. This evening my lounge light bulb exploded. It is a bog-standard fitting, pendant hanging from ceiling with bulb on the end. The metal bit of the bulb was still stuck in the fitting so I had to take everything apart - bits of glass and black plastic everywhere. Now I can't put it all back together again. All the wires are in place so there's no wiring to do. Here's the problem:
inside the dome-shaped white plastic thing hanging from the ceiling is a black plastic fitting. In my hand is the fitting which goes into it. This consists of a metal bit with holes gouged out for the bayonet cap bulb to go into, and a black plastic bit which is meant to fit into the black plastic fitting still hanging from the ceiling. The fitting which is in my hand has two little springs and inside the little springs are 2 tiny strips of metal. Somehow I've got to fit this thing in my hand to the black plastic fitting which is still hanging from the ceiling, but the springs wobble about like crazy and I cannot fit it in. Any suggestions please?