Can bicycling cause runner's knee?
The Wikipedia entry for Iliotibial band syndrome suggests that some of possible causes may be


Inadequate warm-up or cool-down
Excessive up-hill and down-hill running
Positioning the feet "toed-in" to an excessive angle when cycling


This IT Band Pain Stretches, Treatment And Prevention article suggests "Having a properly fitted bike can help triathletes who suffer from ITB impingement while riding", and gives other advice.

Edit: This old answer may be useful also. 

I suspect


because it's such a short ride, maybe you are not warming up enough before you start.
the bike fit, despite your confidence that it's ok.
gearing: if you have to stand on the hills for long periods, then you need lower gears.


I recommend


stop riding now while you find the cause and find the correct treatment.
go to a specialist bike shop and get you, your bike, and your position (including foot position) checked, so that the fit is confirmed to be good.
get lower gears.
get medical advice. I like to see a physiotherapist for this kind of thing. Find one who is really into sports injuries.
depending on the medical advice, do some strength work.
when you resume riding, make sure you warm up thoroughly.
when riding hills, use the lower gears so that you spin more.


Edit: and I should add I am not a medically trained person.