We normally take the mileage deduction. We'd still like to keep the mileage and add depreciation. Is that possible?

No, that's not possible. The standard mileage rate includes an amount for depreciation. You cannot also claim depreciation separately. You would be claiming depreciation twice.

You can change to using the actual expense method, which includes depreciation, but you must track all your expenses, which is a lot more paperwork.  The actual expense method may be more lucrative if your car is new and expensive, but most of the time, the standard mileage rate is more lucrative.  The rules are described in chapter 4 of publication 463,

https://www.irs.gov/forms-pubs/about-publication-463

But you can't mix methods. 

Take a look at https://bradfordtaxinstitute.com/free_resources/irs-mileage-rates.xml.aspx and scroll just a bit past halfway down the page. There you will see how much for each mile is considered depreciation, back to 2003.