The committee launched a bipartisan investigation into the use of "personal email and messaging accounts" by "non-career officials at the White House", but Cummings says in the letter that even after he followed up in December 2018 requesting documents on the cusp of becoming chairman, "the White House failed to produce any additional documents" and "failed to provide the promised briefing during this timeframe".
And Lowell confirmed that Kushner 'had communications with people outside the United States'.
The revelations are particularly embarrassing for President Trump, who during the 2016 campaign called for the jailing of Hillary Clinton over her own use of a secret email account to exchange top-secret information.
Jared Kushner and Ivanka Trump - along with other White House officials - used their personal email accounts and messaging apps for official business, the House Oversight and Reform Committee announced on Thursday.
Lowell would not tell the committee whether Kushner had ever used WhatsApp to discuss classified information.
The dispute arose as Cummings also released information about Kushner's use of WhatsApp and raised questions about personal email accounts used by other former senior White House aides to discuss a proposal to transfer US nuclear power technology to Saudi Arabia.
Previous attempts by the committee to obtain information on the use of private email have been rebuffed by the White House, setting up the latest battle between the Democratic-controlled body and the president.
System Shock 3 gets a very brief teaser trailer
I'm cautiously optimistic and will remain so until we get some more information on the title. But we never explained why she wants to destroy humanity.
Section 2209 of the statue stipulates White House officials are required to forward a "complete copy" of "official business conducted using non-official electronic messaging accounts" to "an official electronic messaging account of the President, Vice President, or covered employee" no later than 20 days after the creation of the communication.
The details about Kushner's use of e-mail and messaging came up because of a new House Oversight Committee demand for documents from Kushner and his wife Ivanka Trump as well as from former Trump strategist Steve Bannon and former Deputy National Security Adviser K.T. McFarland. "The Committee requests that you confirm by March 28, 2019, whether you intend to comply voluntarily with its request or whether the committee should consider alternative means to obtain the information it needs to conduct its investigation", Cummings wrote. In mid-November, Carol Leonnig and Josh Dawsey reported in the Washington Post that she had sent hundreds of e-mails to aides, Trump cabinet officials and her assistant via a private e-mail address she shares with Kushner.
When asked, Lowell said, "That's above my pay grade", and referred questions to the White House and the National Security Council, according to Cummings' letter.
That proposal is under investigation by Cummings' committee, which is looking into information from whistleblowers who have said they witnessed "abnormal acts" within the Trump National Security Council involving senior White House officials who were pushing the plan.
Bannon also received a January 29, 2017, email from Barrack that Cummings said was a pitch of the plan sent to inform "Bannon's official work relating to developing 'broader Middle East policy'".
In a statement, White House Counsel spokesman Steven Groves confirmed the administration received Cummings' letter. "As with all properly authorized oversight requests, the White House will review the letter and will provide a reasonable response in due course".