You ignore months that you didn't have a mortgage. They don't enter into the calculation at all. You can't count them. You calculate the average balance only for the months that you did have a mortgage.
The section that you quoted from Publication 936 applies to using monthly statements from the lender that show the balance each month. Did you get such statements? In your original question you only mentioned a 1098, not monthly statements. If you did not get monthly statements from the lender showing the balance each month, then that whole section under the heading "Statements provided by your lender" does not apply to you. You just ignore it completely.
If you did get monthly statements showing the balance, when the publication says "You can treat the balance as zero for any month the mortgage wasn't secured by your qualified home," it means months that you did have a mortgage, but the mortgage was not secured by your home. But there are no such months. No mortgage lender would give you a mortgage loan without it being secured by your home. So you just ignore that sentence. There are no months for which you can treat the balance as zero.
(I suppose that sentence is there because a private lender, an individual person, might enter into a mortgage that is not secured. But no bank or mortgage company ever would.)
