My commute to work is just about 7km and it sounds like we have a similar climate. My direct commute typically takes about 20 minutes door to door, and that time depends more on the traffic lights than how hard I ride or what bike I take. During the summer (which the OP is in now) it's pretty easy to work up a sweat even such a short ride, so the advice to take your uniform and change at school is a good one, even if you can't take a full on shower. In the evenings, you're going home so who cares if you sweat.