After several months, the best answer I have is my simple tool. It is raw but functional.
What I want is something similar to this. The JSON data can provide a skeleton for the JSON schema. I have not implemented it yet, but it should be possible to give an existing JSON schema as basis, so that the existing JSON schema plus JSON data can generate an updated JSON schema. If no such schema is given as input, completely default values are taken.
This would be very useful in iterative development: the first time the tool is run, the JSON schema is dummy, but it can be refined automatically according to the evolution of the data.