Sofia Legg's family was told there was a six-month waiting list for therapy despite an urgent GP referral for help.
The inquest coroner in Taunton recorded a conclusion the 14-year-old had taken her own life by hanging.
Somerset Partnership NHS Foundation Trust said it had apologised to the family and improvements had been made.
The inquest heard Sofia, a student at Kings of Wessex Academy in Cheddar, had been suffering from low moods, was self-harming and was expressing thoughts about taking her own life.
She was referred to the Child and Adolescent Mental Services (CAMHS) in March 2015, but did not meet the criteria for specialist support.
In February 2016, her GP made an urgent referral to CAMHS, after Sofia confessed to having suicidal thoughts.
But after being seen twice by CAMHS in July 2016, she was told there was a six-month waiting list for cognitive behaviour therapy.
Her family found her body when they returned home on the evening of 20 September.
On the day before she died, a CAMHS worker met Sofia and her mother, giving a crisis plan stating her mother Sandy Legg needed "to watch her and keep a close eye on her".
But investigation reports stated Sofia had to be "supervised at all times".
"If constant supervision was necessary then this should have been stated directly to me before I left with Sofia," her mother told the hearing.
Speaking after the inquest she added: "As a family we will forever mourn the loss of our kind, bright, beautiful daughter.
"Sofia was failed by the system and we, as a family, feel badly let down."
Somerset coroner Tony Williams said he would be following up his concerns about the death with the trust.

The Tinkoff rider finished the 4km time trial ascent of Mont Chery in the French Alps in 11 minutes, 36 seconds.
Australia's Richie Porte, riding for BMC Racing, was six seconds adrift, with Team Sky's Froome third.
Britain's Froome has won the race twice before - in 2013 and 2015 - both times going on to win the Tour de France.
He decided to go out early in the race against the clock because he was concerned it might rain later in the race.
"It was tough. I was running out of legs," he said before Contador and Porte had finished.
"I'm happy with that. Just the feeling makes me happy. I feel like I'm in  good shape. I still have some work to do and I hope to keep the momentum till the Tour de France."
Contador, also a two-time Tour winner (2007 and 2009), is yet to win the eight-stage, week-long Dauphine, but has twice finished second - in 2010 and 2014.
He said: "The climb suited me really well. It's a bit of a surprise to beat Froome and Porte but I knew that I could do well."
Britain's Adam Yates, who rides for Australian-based team Orica GreenEdge, finished eighth, 31 seconds behind Contador.
The three-week Tour de France starts on 2 July at Mont Saint-Michel in Normandy.
Stage one result:
1.  Alberto Contador (Spa/Tinkoff)               11mins 36secs
2.  Richie Porte (Aus/BMC Racing)            +6secs
3.  Chris Froome (GB/Team Sky)                +13secs
4.  Daniel Martin (Ire/Etixx - Quick-Step)     +21secs
5.  Julian Alaphilippe (Fra/Etixx - Quick-Step) +24secs
6.  Wout Poels (Ned/Team Sky)              +25secs
7.  Romain Bardet (Fra/AG2R)                    +29secs
8.  Adam Yates (GB/Orica)                     +31secs
9.  Diego Rosa (Ita/Astana)                      +37secs
10. Jesus Herrada (Spa/Movistar)                 +39secs
Mental health services failed a teenage girl who had suicidal thoughts, her family has said following an inquest into her death.
[NEXT_CONCEPT]
Spain's Alberto Contador opened up a 13-second lead over main rival Chris Froome in winning the first stage of the Criterium du Dauphine.