It is important that we know who is in the country already. How can we attempt to have a sensible policy on immigration if we don't know who is in the country. Many illegals are being forced to work for unscrupulous employers and are living in appaling conditions. An amnesty would stop this instantly and then we could find useful employment for them, they could pay taxes on the earnings they make and they would be released from virtual slavery. As a fine to the employers using illegal workers they could be forced to pay for all their living expenses until the person is settled and in gainful employment. That will stop them bringing more people in again after the amnesty.
The government cannot even give a definitive number to how many illegals are here. Could be tens of thousands, could be hundreds of thousands we don't know.
After the amnesty, if anyone is caught in the country that is not registered then they must have arrived after the amnesty or failed to register for some reason (probably criminals or undesirables) and can be returned to their home country unless there is a good reason for them to stay.