How to keep track of days online? I'd like to display the total number of days that users were online in a View on their profile pages.

How can I do this?
You can use Visitors module.
It will give you the following reports:

View pages that have recently been visited.
View pages that have been hit frequently.
View pages count per month.
View pages count per day of month.
View pages count per day of week.
View pages count per hour.
View pages count per host.
View access log.
View user activity report (hits, nodes, comments).
View referer list.

and many more things.
Maybe for your requirement you need some customization.