While you can normally insert arbitrary unicode characters to any X11 application using Ctrl-Shift-u and four hex digits, it doesn’t work in kwrite or kate. Instead you’d have to press F7 to switch to command line and type in
char <unicode>
For example, to get the degree symbol (Unicode: U+00B0) you’d type in 'char 176'
(176 being 0xB0 converted do decimal).