So far there have been some great replies (bird bath shower, baby wipes, showering before work, etc.). Also the park and ride 1/2 way might be doable or drive into work and ride home might also work.
To me the best piece of equipment you can buy would be puncture resistant tires. After that a rear rack and some sort of basket or panniers will come in extremely handy.
I would still talk to someone about using the showers or locker room at the gym in your basement. Maybe you could rent a locker or something (to keep your clothes or gear in), but not a membership. Otherwise, try to befriend the manager or the usual person who will be there in the morning. They might let you slide.