The toilets[]
In a pinch it will do. At the very least you'll be able to give yourself a quick sponge down and get changed out of your cycling gear; that alone will make a big difference.
Workplace shower[]
Many workplaces have showers accessible to staff. Sometimes they take a bit of hunting out, esepcially in shared buildings, but thanks to new building codes showers are becoming increasingly common in the workplace.
In a shared workplace if you don't get any joy from your office manager, try the building manager. Quite often he'll know things that the office manager doesn't.
Join a gym[]
OK, this isn't a cheap option but it can be the way to go. Not only do you get a shower but you get a place to up the beefcake as well. Even at city prices gym membership is still considerably cheaper than using a car for commuting, however it's probably not cheaper than using public transport.