Suppose you have an 𝑛×𝑛 chess board and a king can attack all adjacent squares, but not the diagonals (i.e. attacks 55 squares including

Suppose you have an 𝑛×𝑛 chess board and a king can attack all adjacent squares, but not the diagonals (i.e. attacks 55 squares including itself). What is the least number of kings necessary to attack every single square and you don't have to worry about the edges being attacked? (i.e. an 𝑛×𝑛 board only needs to have (𝑛−2)×(𝑛−2) squares attacked)

I've been playing around with smaller cases and it seems like you need at most 𝑛^2/5 kings. Is this correct and how would I prove this?