Everything is easier to repair when you have the schematics. I think the ones for the X220 are available around the internet. Get them.
Next, check all power rails. Zeroth rule of troubleshooting: "thou shalt check voltages". Many (if not most) problems can be solved this way.
If the rails are okay, well, then you gotta put your thinking cap on and work your way from whatever signs of life (if any) the motherboard gives, which might involve probing around to find out what is going on (and what isn't). Probing clocks is a reasonable next step.
Keep in mind you can't always win - replacing, say, a toast BGA chip, unless you have the necessary equipment, will probably cost much more than a replacement motherboard.
A perhaps more-to-the-point ThinkPad repair anecdote of mine: I once repaired a T400 that was seemingly dead. The problem was a shorted USB port. IIRC, the USB power chip was toast as well. Replacing both USB port and USB power chip brought the board back to life. No fuses were blown by the short or anything.
Another thing that comes to mind: Some ThinkPads will misbehave when the CMOS battery is depleted. Check that.