Diane borrows $$63,000$ and repays the principal by making $10$ payments at the end of each year into a sinking fund earning an effective rate of $5\mathrm{.}5\%$ per annum, apart from paying the interest on the loan at each year end. The interest rate on the loan is $7\mathrm{.}5\%.$

$($a$)$ Find the total payment that Diane has to make at the end of each year.

$($b$)$ The net interest of the loan at the end of the $k$ th year is $$2,500\mathrm{.}22.$ Find $k.$