Say for example I take a 30 year mortgage and I make repayments fortnightly. How many repayments will I be making, will it be exactly 780? If so, how does one account for the extra days?
It depends entirely how the mortgage is written. Typically, a 30 year loan made with half payments every 2 weeks will pay off early. The higher the interest rate, the shorter the result. It's common to see a projected 23 years or so, payoff.
That said, I can write a mortgage with biweekly payments and calculate the payment to make the term a full 30 years.
Keep in mind, unless the lender is in agreement, they have no obligations to apply small payments every other week, and can set it aside until they have a full payment. In which case you should go back to sending in the payment at the beginning of each month but set aside the money biweekly. Twice a year you will have an extra half payment that you can apply to the principal. This is why biweekly payments pull the mortgage in because you were effectively making an extra month's payment once per year.
how does one account for the extra days?
Like Joe says, it depends on the structure of the loan, however a rough estimate of how much time it will save you is to keep the number of payments the same and just scale the time period.
e.g. You will still make 360 (30*12) payments, but if you make those payments bi-weekly you'll have 26 half-payments (equivalent to 13 full payments) per year instead of 12 full payments, so your time period is:
360 / 13 = 27.7 years
In reality the time frame will be shorter, since you'll be paying off principal faster (due to the extra payment) and less interest will accrue. The best way to calculate it is to create your own amortization schedule in Excel that is customized for your loan structure (will the payments go toward principal, both principal and interest, or be held in waiting until your payment is due?)
We switched to a bi-weekly schedule to align with paychecks, and our time frame went from 30 years to about 26.