The BMW recommended service interval for the air filter is 12,000 miles (20,000 km) which is a joke. The air filters are way oversized and, unless you're in a very dusty environment, will easily last 50,000 miles - which is when I replace them. And I think that's even overkill but it's a nice convenient round number.
The air filter is located on top of the engine on the right hand side.
Here's what the air filter looks like:
Depending upon which model K you have you may need to remove the right side fairing lower or radiator cover to gain access to the air filter.
1 - Remove the air intake snorkel. It's bolted to the radiator by two 8mm hex head bolts. Once those are removed you can just pull it out of the bottom portion of the air box.
2 - Open the three clips holding the top and bottom halves of the air box together. There are two on the rear and one in the front. The front one is pretty well hidden on the four valve bikes but it's there. (Or should be.) These can be "opened" by inserting a screwdriver between the clip and the air box and prying them open.
Air box clip:
3 - Pull the air filter straight out to the side. Both the top and bottom of the air filter have grooves in them that match up to the walls of the top and bottom halves of the air box. However, the top left side of the air filter does not have a "wall" so the trick is to lift the left (or back if you're looking at it from the right) side of the air filter and slide the air filter out to the right of the bike. (If you already have a replacement air filter then take a look at it and you'll see what I'm talking about.)
4 - Install the new air filter. If you look at the sides of the air filter you'll notice an arrow molded into one side of it. When you install the air filter the side with the arrow should be to the rear of the bike and the arrow should be pointed upwards since that's the direction that air flows through the filter.
5 - Replace the three clips, making sure that the grooves on the top and bottom of the air filter are properly aligned with the walls of the top and bottom halves of the air box.
6 - Replace the intake snorkel and bodywork.
7 - Go for a ride....
