My first bike commute was 6.5 miles, so just a little longer. I wore full bike clothes, as the way there had a pretty good climb that produced sweat, even if I went slowly. I showered before work and washed my hair. When I got to work, I used baby wipes and plain old soap and water in the important parts. I did my make up, too. My hair, at that point was wash and go, so I just re-wet it and put some gel in. I changed/cleaned up in the bathroom, which had an outer waiting area and a lockable toilet/sink area. My work clothes were brought to work on Mondays and home on Fridays, when I drove.
Nobody cares what you are wearing and all I got was admiration, especially from the school janitor, who saw me carrying my bike up the stairs.