How do I repair the boat?

First, you need to open the gate.

Once the gate is open, you need to get rid of the water in the boat.

A bucket would be useful.

Unfortunately, you don't have a bucket.

What about some other kind of container?

Like a shoe, for instance.

A wooden shoe, to be more specific.

Use the wooden shoe on the water.

Now you need to plug the leak.

You'll need to get the cork back. (See "I can't find my cork" for details/)

The cork alone won't plug the hole.  It's too small.

You need to pad the cork.

A rag would work nicely.

You need to cut the curtains in the lighthouse with the razor to get a rag.

Use the rag on the leak, then use the cork on the leak.