I agree with some that just using gas can be the quickest and most realistic if you plan on keeping your vehicle. Using the .51/mile reimbursement value can be misleading. It's great if you just want to win an argument or avoid hard math, but when you're using one, catch-all number to apply to everyone and every situation in the country, you know it has to be a very general estimate and may or may not apply to any one person's situation.

Maintenance/wear and tear can be really tricky. If you're strict about changing your oil every X miles, that can be an simple number to add in. Otherwise it's really guesswork and averages. I could go years without a major repair and then get hit with a big one. If the big one comes during the year I was bike-commuting, all of the sudden those maintenance savings aren't there. So figure in costs that have a clear, per-mile cost like oil, gas, and tires. Maybe throw in brakes. Everything else is guesswork, averages, and luck. Some people like to calculate vehicle depreciation, but that's another squishy number. Lower miles keep your vehicle more valuable, but it still depreciates just by getting older. By most calculations, my car depreciated into negative value shortly after I got it, which is ridiculous, so treat those calculation with some skepticism.

Other places to look for savings:
Extra mileage. Sure, you know what your commute is, but do you always go straight to work and back? I don't. If you cut out those other detours, or do them on your bike, you can figure them into your other mileage-based savings. Likewise if you can make other, local trips on you bike, those can figure in. For me it got to the point where I had to be leaving town to even consider using the car, so not only did commuting miles go away, but so did all miles associated with local errands.
Gym membership if you use it and are willing to give it up.
Parking costs at work if you have them.
Insurance costs if they will go down.

Health benefits exist, but are also hard to quantify. Cycling doesn't exempt you from any illness, but makes some issues less likely or less severe. On the other hand, some activity-related injuries might become more likely, so, again, a hard number to pin down. However, regardless of actual dollars spent or not spent, being healthier has real value. So while you might not be able to figure the money to any reasonable degree, I would still not want to downplay the value in general quality-of-life terms.

And, as has been mentioned, there are costs associated with biking. Those can be almost as high or as low as you want them to be. If saving money is the primary goal, bike costs can be minimized, but you might want to figure on some shift of funds from the car to the bike. Of course if you're already riding regularly and already maintaining a bike, then it's possible the increase in bike-related costs would be minimal.

My wife doesn't like to waste money, but she also worries about my health as well as her own. Any money that I put into my bike is money that eventually translates into better health, so she is, for the most part, fine with it. Gas savings are there. Car maintenance savings are there. Medical cost savings are there. Insurance, parking, and no gym membership figure in. But they aren't the justification. They are the gravy. But not the fattening, artery clogging kind.