Proper handling of mileage reimbursement paid by client

While logically it seems to be a reimbursement, it is not, it is income and posts to its own income account if you want, but income and expense are never posted in the same account - not sure if the last comment applies I wasn't real sure if that was your intent or not.

You don't say but I will assume you are using QBO

If you have QBO PLus, In company settings turn on make expenses and items billable
check mark to track billable expenses and items as income
then in company settings>advanced>chart of accounts click the edit pencil and select the billable expense income account

Why intuit separated the two related settings is anyone's guess

When you enter a vendor bill/CC charge/etc, you will see a Billable check mark box, check that and select the customer name in the customer block. The next time you invoice your customer, the system will show billable expenses on the right side that you can click the add link to add the bill to the invoice.  IF you do not want to add the bill to the invoice, leave the billable check mark box blank, the cost of the expense will still count against the customer

Then, on the invoice, add your service item which would point to its own income account.

"I receive a lump sum check that is a combination of fees and mileage."

When you Submit your report to them, you would also List the charges in the QB data file, as an invoice. The two Service items will allow you to put the Mileage rate and use Quantity for the Mileage line item, the same as you submit for X number of Hours with a Rate, typically.

"I would like to split the deposit into my income account and a mileage reimbursement account. "

Don't think of this as happening in the Banking Deposit; the Sale, on an invoice or a sales receipt, already has the Split info on it. The Banking is just your Payment against the data you already have split.