There are several ways to go about it. First way is to ask pepole who work for large companies or school boards what they get paid for driving their own vehicles. One large US company I know pays 38 cents per mile for the first 5000 miles traveled each year. Then it drops to 25 cents per mile. That gives you some idea. So, for 200 school days, you would get 5000X0.38=$1900 plus 13,000X0.25=$3250, for a total of $5150.00, or $25.75 per trip.
Another way is to ask the IRS what mileage they allow as a tax deduction on the tax return. They will have some figure, higher than the company scale, since many self-employed people charge for their vehicles that way.
A third way, which is more subject to dispute, is to take your gas cost per mile, add the projected maintenance and repairs per miles, usually about 7.5-10 cents, add any extra insurance you may have to pay, and allow for the more rapid wear out and depreciation of your car.
A better way is to add up all your current car expenses, including depreciation (make it level, say $15000 over 10 years=$1500/year), insurance, gas, washes, maintenance & repairs, and divide by your miles driven per year to get what it is costing you now per mile. Then charge that, so make sure you don't lose!
Under no circumstances should you accept only "out of pocket" expenses, such as gas and oil, as compensation. You would really be losing out on that method. A quick way to settle any argument will be to go to a AAA office and get their pamphlet on car ownership costs. They have a very high number, but showing it will convince your client how good a deal they are getting.
Whatever you charge will be less than a taxi ride.
There is also the wear and tear on yourself; I would charge the time , say 2 hours and multiply this by what you would normally be expexted to be paid. If that is $15 per hour that's, say $30 for each day.
If you add $25.75 and $30 you should be paid $55.75 per day.
Don't let somone try to low ball you, because there are a lot of hidden cost and risks in transporting kids to school.