Should I charge gas mileage for house sitting for one flat fee or everyday that I am house sitting?

Since you've already communicated that this request is outside your service area, it seems a flat fee, added as a Cost Adjustment, would be the best way to handle this. You may use the comments area to explain the cost adjustment shown is for you to provide services beyond your specified service area, as usually customers would not expect to pay mileage or additional beyond rates shown. When you calculate the fee, keep in mind that you'll receive a percentage of it from Rover, along with the same percent for your services, so you may opt to round up from the IRS calculation (used as a basis) to include the % of the cost adjustment you'll receive.

I would add as a flat fee. Using some formula may just confuse your client and make them less likely to book with you again.