How can I recover after Kate crashes?

Being one of the Kate Developers, I can explain the workflow like this:

When Kate or the system crashes, you loose all text buffers that were never saved. However, if you are working on a text file (that exists as file on disk), a swap file is created next to the file, called .filename.kate-swp. Now, if Kate starts again, Kate searches for these swap files. If found it replicates all edit actions that were recorded in this swap file, and your data is fully restored.

Rule of thumb: Always (and I repeat: alawys) work with files, not unsaved text buffers. That is good practice with everything you do on a Computer.