Some improvement

I created a new category and moved some of the items to it. No result.

Then I started opening and saving one by one. Those show up.

The problem is: I have more then 6000 items...

This indicates that something went really wrong during the migration (captain obvious once again).

I think at this point, you might need to re-run the migration script.
Before you do this however make sure to backup.

I'll give it a try, but I might get to where I am again.

It would be great to have a script automating the open-save-close routine, that would certainly do.

Try the following:
Open phpMyAdmin or an equivalent software.
Go to the #_k2_items table and check an unsaved item's row.
Save it and see which aspects have been changed.
This way you can see which cells and values where affected and you can build your SQL script.

Finally got it!

The field "language" cannot be empty, and all were. After an update k2_items set language = "*" all is good again!

Nice to hear this Carlos :)
Kudos for your hard work!