Ms Wickett said she stayed at the scene and tried to help.
She added: "People are just in very bad shape. The people that were hurt, the ones that could speak, were calling back as we called out to them.
"Lots of people were upset and crying, but there were no screams."
President Barack Obama said in a statement: "Michelle and I were saddened by the terrible accident in north-east Washington DC. Our thoughts and prayers go out to the families and friends affected by this tragedy.
"I want to thank the brave first responders who arrived immediately to save lives."
Washington fire chief Dennis Rubin said approximately 200 firefighters were at the scene of the accident.
He said 76 people were treated at the scene and six of those were sent to hospital with critical injuries.
He said the majority were walking wounded.
On Tuesday morning, Mr Fenty said two of the patients with critical injuries remained in hospital.
The crash had been "the deadliest accident in the history of our Metro train transit system", the mayor said.
The dead driver was named as Jeanice McMillan, 42, by the Associated Press (AP) and the Washington Post.
Survivors of the crash describe the accident
BBC correspondent Richard Lister said it was possible people were still trapped in the lower of the two train carriages.
During the press conference, Mr Rubin said a large crane had been used to separate mangled pieces of wreckage so that rescuers could search for injured or dead.
Mr Rubin had earlier said parts of the lower carriage were 70 to 80% compressed, and that rescuers did not know if there were still more bodies to be found.
A thorough search of the sidings and surrounding woodland had been made, he said.
The general manager of the Washington subway, John Catoe, said the crash had happened as one train waited for another to clear a station ahead.
He said: "The next train came up behind [the waiting train] and, for reasons we do not know, ploughed into the back.
"We are committed to investigate this accident until we determine why this happened and what must be done to ensure it never happens again."
The accident happened at the peak of rush hour on a busy commuter line.
But the trains involved were heading towards the centre of Washington rather than to the city's outlying areas.
This meant they were likely to have had fewer people on them, AP quoted Debbie Hersman of the National Transportation Safety Board (NTSB).
Investigators were trying to find recorders which would have details of the train's speed at the time of the crash and other information which could explain how the accident happened.
She also said the train which had crashed was part of a fleet of older carriages which transport chiefs had recommended be either replaced or retrofitted, AP reported.
Passenger Abra Jeffers, 25, told the AFP news agency: "I was on the train that got hit. I thought it was an explosion. I thought it was like the train bombings in London. There was smoke and dust everywhere."
Fellow passenger Theroza Doshi told the Reuters news agency: "It just happened. There was no slowing down of the train, just a jerk.
"There was no attempt at braking. We just slammed into whatever we slammed into."
The accident is the Metro network's first crash with a passenger fatality since 1982 when three people were killed in a derailment.
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