Engel-Schmidt has been given a three-year driving ban and a fine for being stopped by police while driving under the influence of narcotics in July last year.
The offence was reported by Ekstra Bladet on Tuesday, after the tabloid newspaper was able to confirm details of the incident involving Engel-Schmidt via public documents.
In a statement subsequently posted on his Facebook page, the MP confirmed that he had driven with cocaine in his blood.
"In July last year I made the worst mistake of my life. I was stopped by police because of a faulty light on my car and took a blood test which showed that I shouldn't have been driving because I had illegal drugs in my blood - it was cocaine. That is completely unacceptable in every way," he wrote.
According to a police report of the incident seen by Ekstra Bladet, Engel-Schmidt initially refused to take a police breathalyser and drug test after being stopped, and was therefore arrested, with a blood test later taken instead.
Søren Gade, who chairs the Liberal party's parliamentary group, said on Tuesday evening that he found it "very difficult to see" Engel-Schmidt return to parliamentary duty following the scandal.
"Jakob Engel-Schmidt has done something really stupid and illegal. He says himself it's unforgivable, and I agree with him," Gade said, according to news agency Ritzau.
The MP was already on leave from parliament, having taken on a position at the Niels Brock business college in Copenhagen in October.
The school confirmed on Wednesday that a "severance agreement" had been reached for Engel-Schmidt to leave his post.
The politician is also relatively well known as a television personality, having been a regular guest on TV2's news programme "News & Co.". But he also appears to have lost that job as a result of the driving offence.
"At the moment, this issue and its circumstances cast a shadow over what he is otherwise able to say in a panel discussion," TV2 managing editor Ulla Pors is quoted as saying on the broadcaster's website.
The scandal now looks to have cost the MP a fourth position as the constituency he represents in parliament announced on Wednesday that it would find a new MP to forward its interests.
"We in the constituency have decided to find a new representative for parliament. We cannot accept the behaviour seen from Jakob. This entire story is unacceptable and something we strongly reject," Toke Fredlev, chairperson for the Liberal party in the Lyngby-Taarbæk constituency, told Ritzau.