... why couldn't I find any private key starting with K? ...
private key is just a number. you can see that number in uncompressed form which is starting with 5 (address in middle column) or see it in compressed form (addresses in right column)
if you place your mouse over the private keys (starting with 5 on left column) you can see the numbers: 1, 2, ..... and as you click next page the code simply generates the private keys in base58 on the go based on those numbers: 100, 101, ...
..I tried 5000000..
in other words you wasted your time.