Firing someone who is investigating you is a counterproductive strategy.

The one thing you never want to do in politics is take the hit for someone else's misconduct - especially someone as erratic as President Trump.

It's generally not wise to pick a fight with law enforcement in politics, especially when you're in the middle of a pretty significant scandal already.

FBI directors serve at the pleasure of the president, but they often have terms that transcend partisan transfers of power at the White House. Firing an FBI director is a major problem from the point of view of a president who is already facing significant questions about investigations already underway.

Presidents are given immense power in the American political system when it comes to personnel. FBI directors are given ten-year terms to insulate them from politics as much as one can, but it's not absolute protection. If a president wants to fire an FBI director, they can do that.