All right, so Neblix helped me set it up, big thanks to him!
I was just following instructions, but here's what we did:
1. In FL Studio add Kontakt channel (Channels->Add one->Kontakt). I you don't see Kontakt on the list, go to More... in the same menu on top of the list instead, find Kontakt on the list and check them out. You should be able to see it in the "Add one" list now.
2. Kontakt will open in a Fruity Wrapper window (may take some time before it opens).
3. Click on the small gear button at the top left of that window ("Wrapping settings"). It will switch to a settings view. In settings tab, set Input port and Output port to 0 and 1, respectively. Go to processing tab and click auto map outputs.
4. Click on the gear button again to go back to main kontakt window.
5. In the middle of the window, there will be an "Outputs" area (the one with sliders). Click Add Channels there and add as many channels as you need. New sliders should appear, named "st.1", "st. 2", etc.
6. For each of those, below the sliders you will see for st. 1 a button with "1|2" on it. Others will have "-|-" on them. Click the first "-|-" button. A "channel output configuration" window will pop up.
7. Click on "not connected" lines in that window and choose outputs matching the name of the channel, so e.g. for channel name st. 2 choose Kt. st 2 [1] and Kt. st. 2 [2]. Click OK. Repeat for all the "-|-" buttons.
8. Add some instruments to the middle window (Multi Rack) in Kontakt main window. I just dragged a few there from the list on the left. When you look at the added windows, they will say "Output: st. 1", "Midi Ch. [A] 1" for the first one, "Output: st. 2", "Midi Ch. [A] 2" for the second, etc. You might need to change "st." for new instruments.
9. Add a new "MIDI out" channel to the Pattern window in the main FL Studio view. Left-click on it in the Pattern window and make sure the channel is set to the instrument you want for that channel (compare to channel number on each instrument in Kontakt window).
10. Click on "Kontakt 5" in the step sequencer again. In "channel settings" (the smaller window, not Kontakt window, the Kontakt window may be covering it up), in top right, there is going to be FX control. Set it to 1.
11. Now if you play on the mini-keyboard for each channel in FL Studio, you should hear the correct instrument. If you look at the Mixer, you will see the Insert 1 play for the first channel, if you switch to another channel, the Insert corresponding to it should show output. If not, make sure you selected FX for main Kontakt channel, assigned all instruments to separate "st. n" and auto-mapped outputs.