Guys,
I resolved my problem - partialy. First of all, instead of 10kOhm resistors, I user a voltage splitter using 150 and 220 Ohms.
After that it still didn't work - for my own sake, I tried software SPI, and it worked. I don't know why, there is completely nothing on my pin 13 output (clk), probably there is something wrong with my MCU... Will try it with another Arduino and check if it works.
Thanks for help everyone!