The reason you are likely having problems finding the replacement instructions for the air filter on your Kenmore Elite Refrigerator is even though that refrigerator carries the Kenmore name brand it is actually built by LG. So the filter is an LG Replacement filter. To replace it first remove the old filter by turning the filter cover to the left and it will pull out, the old filter is mounted inside the back of the filter cover. Just pop it out and insert the new filter and mount the cover back on the refrigerator. The new air filter can be order from the link below if you do not have one yet.
Kenmore Elite Refrigerator Replacement Air Filter