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In order for the starter gear to engage the starter must turn fast enough. If the shaft and gear were rusty and dirty then chances are the inside of the starter is the same way.
Make sure the wires you are using to jump the starter are large enough to carry the load of the starter.
Any automotive alternator/ starter shop can rebuild it or another alternative is an aftermarket starter.
The starter from Honda is almost $650! I wouldn't mess with it I would recommend replacing it.