I am a physician (not practicing anymore), a biker, and an office worker. And I ride fixie. And I sometimes have knee discomfort. So, my few cents:
- Depending on HOW YOU SIT DURING WORK, specially how you rest your legs/feet, your knee might be twisted for a long time (hours a day). Watch yourself if it is happening or not.
- Take a rest on the fixie. I noticed it is only nice when you are "energized". Sometimes we are a bit tired or "low mood". These are not good times to ride fixed, I think. So, take an abundantly-geared beater or something, until you get better.
- If you don't get the rest, try not to accelerate or decelerate too fast. If you ride in traffic, give up riding faster than motor vehicles, for a while, until your knee gets better. I think speed is not so important (spinning is winning, Stephen said), but bursts of acceleration or deceleration put a stress on the knees.
- Watch yourself while pedalling. If you notice some movements reproduce the pain you feel, change your habit, avoid them as much as you can. But try to keep efficient. If you have to create an anti-natural position so as not to feel pain, things are starting to get ominous!
Other factor: you mentioned your cycle is short, less than 15min (so is mine). I found this to be a problem, because you always ride "cold". Perhaps you could warm-up and stretch, at home (learn how to stretch thigh and calf properly, first), or even take a LONGER way to work, so you can ride lightly for more time, thus pumping some nutrients inside your joints' tissues instead of just hammering'em cold.
At last, seeing a doctor (preferrably specialized in sports medicine) and doing some physical therapy is always something to consider. But don't think that is more important than self-observation, self-knowledge and self-control.
Hope it helps.