Basic 12V Step-Start Circuit For Automotive Fans: Troubleshooting
The circuit cannot work-220K is too high for that transistor due to internal resistors. D2,3 do no good- they should be across coils of K3,5 (reverse biased, or you'll fry the transistors).

You could try replacing the transistors with power n-channel MOSFETs or greatly reduce the resistors and increase the capacitors proportionally to compensate.
Something more like 47K and 470uF, 10V rating is okay for the caps but get 105C types. (Note: This was assuming the original circuit was calculated correctly, but I don't think so). 

Edit: See below circuit. R2 represents the relay coil (80\$\Omega\$ at 25°C), this will give roughly a 1.7 second delay with 47K and 100uF. Change R1 if you want more or less time. I have used an IRFZ44 MOSFET in place of the Darlington. 







simulate this circuit &ndash; Schematic created using CircuitLab