I firmly believe if we secure the border and enfoce immigration laws already on the books, and deny them access to social programs, many would deport them selves. Most of the illegals were employed in Mexico when they left, many have homes and families there that they send money to. Those who have roots in Mexico would return if we made it difficult for them to be here.
We could deal with those that remained as we bacame aware of them. Any arrested for other reasons could be deported as they are identified as illegal. I disagree that it can't be done. I think people who promote amnesty want people to think of a large operation to round up and deport the illegals all at once. It would not have to be done that way at all. It would work if it was done in stages.
Once the population of illegals was down to a more managable level, we could discuss the need (or not) of foreign workers and then set up a program for legal workers if needed.