Remove the Handle
Normally the handle will break where it meets the spade but the wood is normally riveted to the handle. To get to the wood this rivet will need to be removed. The easiest way to accomplish this is to drill into the head of the rivet and it is a good idea to use a hammer and center punch to make a small indentation to stop the drill sliding everywhere. Now just use a punch to knock the rivet out. Now you can use a flat bit to get all the wood out just make sure the wood has dried first.
Adding a Handle
Once the hole is cleared you want to insert the new handle. Spade handles are pretty standard and all you need to do is line it up and then insert it into the hole. Give the spade a few sharp taps on a solid surface so the handle fights tightly. Now you need to secure it with a rivet which is easily improvised. Drill a hole through the handle using the previous rivet hole as guide for size and push a nail through so about a quarter of an inch protrudes from the other side. Now place the whole thing on a hard surface so you can tap the head of the nail with a hammer to form a rivet.