A nation that cannot protect its borders is not a nation. The primary responsibility of our government is to ensure our freedoms, followed closely by keeping us safe. Paramount to meeting the minimum requirements of such would be security over who and/or what comes into our country. Until they can secure the boarders everything else should take a back seat (now I know some might point to things they consider more important but I will insist that since securing the boarders is something required and something that can absolutely be done that it would be a priority over others).
What to do with those illegals already here? I do have some compassion and would say if they have been here for more than 15 years and properly assimilated and have not been in any trouble and have maintained a good work record that they should be allowed to stay. If they still hold to their old nationality and strictly speak its language and still find themselves cheering for that country over disputes of both governments and sports and any other such representation, that they should leave since their homeland isn't here by their implied admittance.
I wouldn't let a burglar stay in my home just because he was able to break in, even if he did mow my lawn. If it was a family, though, that was in desperate times and one that felt they had no other option, I still wouldn't, but I would likely try to find a way to help them. Our government should hold our opinions above those of the rest of the world and it should do what is in our interests and not the interests of foreign nationals -- regardless of their plight. Their can be exceptions though, and even though these invaders demonstrate a lack of loyalty for their original countries and that they are the types that would stay only as long as the going is good and easy, it doesn't necessarily mean that their children are. If all their children have ever known is our nation then we should give them consideration, even if that consideration might include their dubious parents.