Retractor springs can quite often be a bit fiddly. I did one a few weeks ago on my girlfriends' brush cutter, but the theory is the same.
You need to wind the spring into its housing whilst maintaining a good amount of tension on the spring. Once you've done this you will need to place the retaining plate back on and then screw it back in. Once the spring is properly in its own housing then you need to place it into the cord puller (you may need to wind it in??? depends on the type) and then re-assemble your cord puller housing and fit it to the mower.
One little warning, make sure you wind everything in the right direction, it's no fun to have to pull it all apart again because you've wound the spring the wrong way into the housing, this I know from experience