Now that you have the Bracket apart, inspect the Plunger for wear. If it's not smooth and shiny, replace it. If the end is mushroomed, replace the Plunger and the Bracket; as it's probably too worn to fool with. If the Plunger has become magnetized and is covered with metal shavings, replace it.
Clean any parts you are keeping with Fantastik or any other degreaser using an old toothbrush. Clean inside the Coil too.
Replace the Sleeve. If it's stuck tight in the coil, press the protruding part down flat on your workbench and push it out. If it is hopelessly stuck, replace the coil (it probably overheated at some point, and could be a liability latter on). Remember that the lip on the Sleeve goes on the diode end of the coil.
The Bracket has 2 yokes. The metal one is almost always broken, the fiber one is always worn. Always replace them both.
Make sure both screws are the type with lock washers installed on them. If the Yoke Retaining Bracket comes loose, the Plunger will become damaged.
When reassembling the Bracket, remember to press the Yoke Retaining Bracket towards the Coil Stop end of the Bracket as you tighten the screws. By doing this you will ensure that there is no slop in the Coil; and thus all the energy will be transmitted to the ball. The Coil should NOT be loose or have any play.