SD card reader does not work

Since it doesn't work in Linux, the problem is likely caused by the motherboard. Depending on if the chipset or reader is damaged, you may be better off writing the internal reader off and using an external one. On the other hand if it's simple you may be able to recover the internal reader and use it again.

To find out who makes the SD card reader chipset, look at the chipset near the card reader. For many years, Lenovo has used Ricoh parts for this but this may be different on the X220 then what my T420 uses (more then likely, it's the same to reduce cost). The easiest way to confirm this is an entry that says Ricoh Co Ltd PCIe SDXC/MMC Host Controller entry with a lspci -nn probe (or something similar). To get a better idea if you just get that, run sudo su and then lspci -vv. If any better information is available, this will provide it. On mine, it's still vague (even if I run the -vv option).

Once you know what chipset it uses just in case you need it (and where it is), check for obvious board damage (burned capacitors, resistors, damaged slot, etc). If you see any of this, the chipset may not be at fault and the issue may be repairable. However, this can damage the chipset if it causes a short bad enough to kill the chipset, but not disable the entire system. If you want to see if the reader can be recovered, try to repair it and see what happens. If it doesn't work after a repair and everything checks out, the chipset is also damaged.

If the chipset is damaged, this is difficult (and uneconomical) to replace. An external reader makes more sense at that point.