If the solenoid is working and the starter motor spins, then it's the internals that are broken. The simplest way to see if it's a cheap fix, is to check if there's play in the arm that is actuated by the solenoid. Outside of replacement of the bendix clutches and helical shaft mechanisms, there's no "real fix".
If you can see the starter's gear teeth in this engine, you could pry up the starter shaft and wrap some twine to "pre-engage" the starter's teeth into the flywheel teeth.
Wrapping the twine in the opposite direction of the starter's turning would assure that it would unwrap or disintegrate when the engine started. Any other method, including plastic and plastic cable ties, might result in damage to persons or engine. Make sure the starter is driven only a brief moment by the engine, should it start. The starter could come apart.
But for this one brief event and traveling such a short distance, why not borrow a friend's boat?
Or, dare I say it, borrow kayaks?
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