Describing the outcome as an "historic trial" in a statement through the deaths in custody charity Inquest, the family said it was relieved that everyone now knew the truth of how he died.
They also called for improvements to the system for prosecutions involving police defendants to prevent delays in future.
Monk was found guilty of the manslaughter of Mr Atkinson but cleared of his murder after a trial at Birmingham Crown Court.
In a statement Mr Atkinson's family said: “Dalian Atkinson is much missed by all his family and friends and the footballing communities of the clubs he played for in his long and successful career as a professional footballer, especially Ipswich Town, Sheffield Wednesday and Aston Villa.
"The past five years have been an ordeal for Dalian’s family.
"We knew years ago about the terrible injuries inflicted by PcMonk on Dalian, but have been unable to talk about them due to the criminal process. We are hugely relieved that the whole country now knows the truth about how Dalian died.
"While it has been hard for us not to be able to talk about the details of Dalian’s death, it has been even harder to sit through this trial and to hear PC Monk try to justify the force he used.
"On the night he died, Dalian was vulnerable and unwell and needed medical attention. He instead received violence, and died with Pc Monk’s boot lace prints bruised onto his forehead.
"We have been sickened to hear Pc Monk try to minimise the force he used on Dalian and exaggerate the threat he posed. Fortunately, the jury has seen through the lies and the pretence. We would like to thank the jury members for all their hard work and attention.
"The fact that this case has taken nearly five years to get to trial is completely unacceptable, especially when you consider that the Pc Monk’s identity was known to the prosecuting authorities from day one.
"By contrast, the murderer of George Floyd was convicted less than a year after his death. Our system for prosecuting police officers must work better in future to get rid of these unjustifiable delays. No more excuses – no more delays.
"Dalian’s footballing talent led him to achieve great things in his life. Our sincere hope is that now that the truth about his death is known, and justice has been done, we can start to remember him not for the manner in which he died, but for the way in which he lived.”
The jury trying West Mercia Pc Mary Ellen Bettley-Smith, who denies assaulting Mr Atkinson, has been sent home and will resume its deliberations at 10am on Thursday.