Amy El-Keria was found hanged in her room at the Priory Hospital, Ticehurst, after tying a scarf around her neck.
The inquest in Horsham heard staff were not trained in resuscitation and did not call 999 quickly enough.
The jury said Amy died of unintended consequences of a deliberate act, contributed to by neglect.
It said staffing levels were inadequate, and a lack of one-to-one time caused or contributed to the teenager's death in November 2012 in a "significant" way.
Amy had a complex range of problems and mental health diagnoses, and died within three months of being moved to the Priory after being asked to leave her specialist boarding school in Berkshire.
The inquest heard that while at school she had drawn a picture of herself hanging with the words: "If only this could happen, but I haven't got the guts."
She made several attempts to end her life in early 2012, and was deemed high-risk when she was admitted to the hospital.
The inquest was told hospital staff had not always been able to give the teenager one-to-one time due to pressure on wards.
Jurors found that risk assessments were not properly carried out, staff did not assess the risk of Amy being able to take her own life in her room and opportunities were missed to remove the scarf from her.
A delay in checking on her on the evening she died also contributed significantly to her death.
Speaking after the inquest, Amy's mother, Tania El-Keria, said: "For 14 years we kept Amy safe. In less than three months under the care of the Priory she was dead.
"If I had treated Amy and neglected her needs in the way Ticehurst House did, she would have been taken away from me."
Dr Sylvia Tang, Priory Group medical director, said: "We would like to offer an unreserved apology and our heartfelt sympathies to Amy's family.
"Following the incident, we undertook an extensive investigation and strengthened a number of our procedures at the hospital.
"We will now review the findings of the inquest very carefully and consider whether further improvements can be made including in relation to staffing, care plans and risk assessments," she said.

The will was filed at Manhattan Surrogate court in New York earlier this week leaving instructions from the pioneering rapper, also known as MCA.
It read: "Notwithstanding anything to the contrary, in no event may my image or name or any music or any artistic property created by me be used for advertising purposes."
Yauch died of cancer in May aged 47.
His will leaves his $6.4m (Â£4.1m) estate to his widow, Dechen Yauch, and their 13-year old daughter, and gives Dechen the right to sell and manage his artistic property as the executor of the estate.
The phrase "or any music or any artistic property created by me" was added in handwriting, according to Rolling Stone magazine which has seen a copy of the document.
Yauch founded the Beastie Boys with rappers Ad-Rock and Mike D in the 1980s, going on to have a string of hits including Fight for Your Right (to Party) and Sabotage.
He was diagnosed with cancer of the salivary gland in 2009 and underwent surgery and radiation therapy, which delayed the release of their latest album Hot Sauce Committee Part Two.
Corporations have regularly used the images of deceased musicians, celebrities and historical figures in their advertisements, such as Apple's 1997 Think Different campaign featuring John Lennon, Martin Luther King Jr and Gandhi.
This week the remaining members of the Beastie Boys took legal action against US energy drink company Monster over allegations of copyright infringement.
They lodged papers in Manhattan federal court on Wednesday, claiming executives have been using their tracks in a promotional video and downloadable audio file without their permission.
The death of a 14-year-old girl may have been prevented if she had received proper care at a Sussex mental health hospital, according to an inquest jury.
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Late Beastie Boys member Adam Yauch has used his will to stop people from using his music or image in advertising.