Jordan,
This can totally be done, and five miles is no problem. Sure, a shower would be nice but it's not necessary.
I will share some details of how I've done it but they might be unnecessary. The main ideas that you can use and adapt to your situation are:
(1) It can be done.
(2) You can transport your work clothing if desired/necessary -- even in bad weather.
(3) You can leave your work clothing at work if desired/necessary.
(4) You don't have to ride that intensely.
(5) There are numerous ways to eliminate sweat/odor once you get to a bathroom.
There are a million different ways to do this and you don't need to do the same every day. Also, you'll just get a sense of what it accepted/allowed at work.
I've done it in multiple environments, including very corporate.
If it wasn't too hot or cold, sometimes I would bike in wearing my suit pants, dress shirt, and tie, along with tennis shoes. In my pannier I'd have my suit jacket loosely rolled and a change of shoes and socks. Upon arrival in the parking garage I'd change my shoes and socks and put on the jacket and go up. Or I could change shoes in a bathroom inside or someplace else unobtrusive. In the beginning I tried to always enter in full professional dress. After a while I chilled out and didn't mind changing shoes at my desk etc.
If it was hot, I'd ride in non-work clothes with my entire change in panniers. I'd get to the first bathroom. Wait a few minutes to stop actually sweating, then wipe down with wipes or wet and/or soapy paper towels. A single occupancy bathroom is ideal but you could also bring a few wet paper towels and deodorant into a stall, if you think being seen will cause a problem.
In many cases, I start worrying there will be some issue if they see me wiping down in the bathroom and after a while I realize (A) I do it in a low-traffic bathroom so hardly anyone sees, and (B) nobody cares anyway, really, if anything they are kind of impressed that I ride.
In a perfect world your ride to work is more downhill and the ride home is more up. Though flat both ways is nice.
And after a while I usually end up keeping my suit jackets and dress shoes at my desk instead of carrying them back and forth every day.
Sometimes I would wear the tie the whole way, sometimes I would put it on at home then loosen it and slip it off over my head, and slip it back on at work (these first two if I was trying to be fully professionally dressed before entering the building). But most often just tie the whole thing at work in the bathroom.
I have all the normal raingear like rain jacket and rain pants but for very hot weather I might just wear a shirt, bike shorts, and sandals in the rain.
For me not actually owning a car was helpful because it forced me to be creative and find a way. I'm sure there are days I would have wimped out if I could have just conveniently driven a car, but instead I found a way to bike it and it turned out fine and I was glad I did.
(Regarding type of bike... I've always done my work commuting in an urban area on either a hybrid or a folding bike. Both are very upright and work well, so I guess I could recommend those types, though I haven't tried anything else so can't knowledgeably compare.)
Have fun!
