You should not need the Power Extender Kit for this install. The thermostat wire that has the green, white, red, and blue wire will also have a yellow wire in it. I think I even see the nub of cut yellow wire peeking out of the brown cable jacket. To simplify the installation, I would try and use that yellow wire.
Trim down the cable insulation on both ends and find the yellow wire. Then connect the yellow wire to the Y terminal on the air handler (along with the red that was there). Then connect the blue wire to the C terminal (along with the white wire that was there). Once that is done, then connecting the thermostat will just be matching colors to the correct letters: blue to C, yellow to Y, white to W, green to G, red to R.
If for some reason the yellow wire isn't usable because you can't access it on the thermostat side, you can make this work with the PEK. The red wire that was connected to the Y terminal needs to be put back under the Y screw. This is the signal cable that goes to your outside compressor. Right now the compressor isn't running because that red wire isn't where it is supposed to be.