Well, darlingtons have 2 transistors & they cannot both saturate so the VCE will be kinda high. The important info from the datasheet is shown in the attached image. So, the VBE is 2.5V when on, and the HFE is >750 A/A. So if the 555 output voltage is 8V and you want at least 4A to flow, the resistor value would be found from (8 - 2.5)/R*750 = 4, or R = (8-2.5)*750/2 = 4k; a resistor value less than that should be fine. VCE is going to be >3V @ 4A.
You might want to have a look at using a MOSFET instead as it's not hard to find one that will have less voltage drop across it in your application.