The leasing company shouldn't ask for your business miles, HR should. You should report that to them, and hopefully they will only include the personal miles on your W-2. However, if you have not been regularly reporting your business miles, they may not be allowed to.
No, personal miles are not deductible on your tax return. However, BUSINESS miles sort-of are. If HR is unwilling or unable to remove the business miles from your W-2, you can enter the "Job Expenses" into TurboTax. Unfortunately, that is part of a category that is limited to the amount over 2% of your income. For example, if your income on your tax return is $50,000, the first $1000 won't be deductible. If you don't usually Itemize your deductions (because the Standard Deduction is larger), the mileage deduction would further be reduced or eliminated.
As for the fact that the lease is not in your name, that is a bit tricky. However, because you are essentially being charged for it (by adding to your W-2), my first thought is to go ahead and claim the expenses as Job Expenses anyways using the "Actual Expenses".
