The keyword you're looking for is "Hardware Maintenance Manual". Lenovo provides extremely detailed instructions as well as replacement part numbers, called Field Replaceable Units (FRUs), for every component of every system they sell. All of this information is within the HMM for the product.
here is the HMM for the X220 and X220i.
If there is no special information about how to replace the NIC, then the NIC is not a separate part and is permanently soldered into the motherboard. If this is the case, then only a full motherboard replacement will be able to replace this card, unless you have extremely detailed knowledge about sensitive microelectronics manual diagnosis and repair. Such knowledge would also require very specialized tools to be useful, and you'd need to find a detailed electrical diagram of the NIC. I don't even have the knowledge to help you along that path; all I know is that being able to solder something is only the very tip of the iceberg as far as trying to "repair" an existing part without fully replacing it.
You may want to verify that it is truly a hardware problem, rather than a software problem, by booting up a Linux live CD and trying the NIC for a few hours. If the behavior is the same on Linux, it's definitely hardware.