To put your decorative paper punch back together, first ensure all the working surfaces are clean, and treat the surfaces with a product like "cutter glide" (available online or from craft stores) to help keep the blades running smoothly. Test that the blade fits smoothly back into the base, you may have to rotate some patterns to get the correct fit. The blade should easily pass through the base, and slide easily back up again.
If the blade fits correctly into the base, remove the cutting blade from the base and refit the spring, then the blade. (see the photo on page 3 of these instructions). Align the plastic cover that fits over the blade with any alignment marks (see photo this page). This will ensure you know which way your design lines up when you go to punch your paper. If possible (most punches will allow this) run a test of the cutting surfaces on a piece of paper, taking care to keep your fingers well away from the cutting blades and the spring. Finally, when you are sure your punch is functioning well, refit the cover, and press it down over the tab that holds it in place.
Hopefully your punch will give you many more years of service (just avoid whichever papers got you stuck and in need of repairs!).