At that budget, Craigslist FTW.

Best candidate, look for a 90's era mountain bike. Preferably old enough to have a rigid fork instead of suspended, but not so old it has cantilever brakes, V-brakes are so much easier to deal with.

Bonus points if it already has slick/urban tires instead of knobbies and a rack.

You could also do well with pretty much any 'hybrid' bike. 5-6mi is a moderate distance, depending on your fitness (which will improve) and hills, you'll probably be doing 30-60 min each way.