The Gate-Source pullup resistors, 5K should work well. Provides 1mA of pullup curent.
The Arduino-Gate resistor, 220 ohm , will pull the gate good & low. Vg will be ~5V*220/(5000+220) = 0.2V.
The LED current limit resistor:
(5V - Vce-pnp transistor - Vf-led - Vds-shift register)/(current-desired) = R
So an example might be:
(5 - 0.5 - 3.3 - 0.08)/20mA = 56 ohm
0.5 & 3.3 are my speculation, dependent on the parts selected.
0.08 is from Rds of TCIP6B595 of 4 ohm at 100mA, so 4ohm * 20mA = 0.08V
So read your datasheets, plug in your numbers & desired currents, go from there.