I'm not too far from you (Milpitas), and my commute is similar...well, when I add some climbs in the foothills at the end of my commute.

I was commuting on a Rockhopper, and it worked just fine, but I wanted a dedicated commute bike. I found my 2012 Trek FX 7.5 on Craigslist for $400. Talked the seller down to $350. Picked up some decent lights, a rack and trunk, fenders, and I was all set for commuting for under $600.

I, also, don't have a place to shower at work. I find that showering before I leave, and then a quick wipe-down with a wet wash cloth in the restroom is sufficient.