Electronic configuration of Potassium (K) :

1s²2s²2p⁶3s²3p⁶4s¹

After first ionisation enthalpy its configuration becomes :

1s²2s²2p⁶3s²3p⁶

which is a inert gas configuration. To remove an electron from inert gas configuration requires very high energy.

So Potassium (K) have high difference in the first ionisation and the second ionisation energy as it achieve stable noble gas configuration after first ionisation.