There is an easier way.
Before we get into that....you have checked the fuse on the engine wiring harness haven't you?
Okay then.
First off, am assuming you have a control box and key switch controlling the engine.
Turn the key to the "Start" position and move the shift lever back and forth a little bit.
What this will do is allow the internal neutral safety switch to make contact if it is out of position.
Try that first. If that works it is an indication you need to adjust your shift cable. More on that later if you need to do that.
Second, if that does not work then you can take a small ga. wire jumper and go from the battery lead on the starter solenoid and touch the yellow/red terminal on the solenoid. That should engage the starter (if the solenoid is good).
Try to avoid tapping on the starter. There is a chance that the starter is not the original and if that is the case the replacement has permanent mold magnets which are only glued to the starter housing. Tapping will loosen them and or crack them very easily.
The only reason a starter reacts to tapping is the possibility of a dirty contact on the brush or brushes. To avoid tapping you can simply turn the starter several revolutions by hand and "clean" the contact points.