You have four things to scan in total:
Scan the Archon (battlewagon) first. Just target it and get close enough for the scan lines to do their magic :) Don't bother with its subsystems.
Now go back to the original mission target - the asteroid base. Target it by selecting it with Target Escort (default "e"). I hope you remember what key you mapped to Target Next Subsystem... Keep hitting it until you see the name of the subsystem in the target box. You need to select one of the three mission targets (Weapons, Sensors, Comm IIRC), fly to it (your HUD's targetting will highlight the selected subsystem of the target ship if its in visual range) and try to get close and get it centred to bring up the scanlines. This can be tricky on large ships due to things sticking out of them, and this here Asteroid Base is no exception. Oh, and...
Once you've scanned the first target subsystem get your skates on and scan the other two ASAP - 'cuz merry hell is about to break loose...
You'll know if you've scanned all the right targets because you'll get an update and your Escort List will switch to show a position marker instead of the base. NOW run for your life (toward the marker preferably!) - divert all power to engines, max the throttle and abuse your burners. Once you reach the marker you'll get a repeating message, hold tight and wait for your ride home.