Jared Kushner was unfazed by the audio recording in which Donald Trump could be heard bragging about grabbing women's genitals without consent.
Kelly said the "intense personal ambition" of Trump White House staffers, who backstabbed and leaked to the press, was "out of hand."
Rep. Elijah Cummings and the House Oversight Committee are investigating violations of the Presidential Records Act in relation to Kushner's work.
Trump's former personal lawyer Michael Cohen had recently testified that the president directed him to threatened schools not to release information.
According to new White House tell-all, Ivanka Trump frequently asked to travel on Air Force planes when it was not always necessary.
Trump previously denied playing a role in the process and said he was "never involved with the security."