Never go against the family.
It's not just a maxim of the Corleones. It is also how Donald Trump runs his businesses. And what bigger business does he have right now than the United States of America?
So there was no reason to think that he would fire Jared Kushner, his son-in-law, no matter how much Kushner's various ethical entanglements and conflicts might hurt the administration. Even if the president's own lawyers were pushing for it, as The Wall Street Journal reports:
Among their concerns was that Mr. Kushner was the adviser closest to the president who had the most dealings with Russian officials and businesspeople during the campaign and transition, some of which are currently being examined by federal investigators and congressional oversight panels. Mr. Kushner, Mr. Trump’s son-in-law and confidant, has said he had four such meetings or interactions. [...]
Mr. Trump wasn’t persuaded that Mr. Kushner needed to leave. One person said Mr. Trump’s view was that Mr. Kushner hadn’t done anything wrong and that there was no need for him to step down.
Trump said the same thing about his first National Security Advisor, Michael Flynn. That did not work out so well for him, and his continued belief in Flynn caused problems, even after he resigned.
Kushner has been sort of a Trump whisperer for the president, and at one point was reportedly often the last person Trump spoke to at night. His status as a family member has led to a wide perception among White House watchers that he is untouchable, even as other high-level aides have quit or been fired in the past eight months.
And yet, his entanglement in the Russia investigation might very well cause him enormous legal problems, which means more pressure on Trump. His enemies in the White House might have been unsuccessful in pushing him out this summer, but that does not mean they were wrong about his legal exposure.
Fortunately for Kushner, his father-in-law is often the last to see a problem.