What exactly does the term 'failed' asylum seeker really mean? Put simply it means that a person [the asylum seeker] has applied for asylum i.e. to be accepted as a refugee under the terms of the 1951 UN Convention Relating to the Status of Refugees*, and had their application turned down under the country's own self-regulated criteria.
Historically, the movement of peoples across borders was always a small-scale event until the late C20th, at least in terms that rich First World countries would find economically threatening. And the mass movements of populations across Third World borders as the result of war or famine never posed such a threat until the advent of cheap air travel and the globalised economy.
It was only then that what we now understand to be modern immigration controls came in as before then international travel was the privilege of the wealthy few plus the armies and bureaucrats that carried out their delegated activities world wide c.f. the British Empire, Dutch East India Company, etc.
The first immigration controls in the UK didn't occur until 1905 and the Aliens Act, which was specifically targeted at limiting Eastern European Jews from entering the county and as a direct result of the proto-fascist agitation of the British Brothers League in the East End of London. However, it wasn't until the 1962 Commonwealth Immigrants Act and 1968 Commonwealth Immigration Act were passed, and the right to settle in the UK was removed from the citizens of ex-Empire countries, that recognisably modern immigration controls arrived in the UK.
Again this was a direct result of fascist and racist agitation, this time by Oswald Mosley's Union Movement and Colin Jordan's White Defence League, which precipitated the 1958 Nottingham and Notting Hill 'race' riots. Both Immigration Acts reneged on the promise to those Empire soldiers who had fought for King and Empire, a promise guaranteed by possession of a British passport, that, as subjects of the Empire, they could settle in Britain after the war if they fought to defend it. Then, when they and their families could finally afford to avail themselves of that promise, the opportunity was snatched away from them.
The 1962 and '68 Immigration Acts were then consolidated by the 1971 Immigration Act, which brought in a provision for the 'administrative detention' for an 'indefinite period' of those who have been refused leave to enter the UK or who are required to submit to further examination at ports of entry; internment in other words.
Subsequent Acts have sought to codify the criteria under which the UK 'seeks' to carry out its international obligations under the UN Convention, and the other treaties it is a signatory of, with regard to refugees and asylum. Inevitably, those subsequent Acts have sought to further restrict the 'rights' of asylum seekers to gain meaningful refugee status, both in terms of numbers and length (duration) of the protection granted.
And to the general dismay of the political class, those restrictions have failed to keep up with the ability of people to afford, both in money terms and in terms of desire/desperation to reach the UK to build a better life for themselves. Hence the massive increase in anti-immigrant rhetoric and the ever spiralling attempts to legislate against migrants in general and asylum seekers in particular.
So, the bottom line is that there is no such thing as a 'failed' asylum seeker, just more and more REFUSED asylum seekers - refugees who have failed to jump through the ever increasing number of hoops that have been placed in their way before they can gain any form of meaningful protection under national and international laws.
* "Owing to a well-founded fear of being persecuted for reasons of race, religion, nationality, membership of a particular social group, or political opinion, is outside the country of their nationality, and is unable to or, owing to such fear, is unwilling to avail him/herself of the protection of that country."