``Why should they be allowed to stay?``

I can think of one reason: the US economy.  The illegal immigrant population is so large now, it has a huge--and currently, very positive--effect on the economy.  Why mess with that?  The illegals impact the economy in a number of significant ways: 

1. 65% of farm labour (as of 2009--source, Food Inc.) are illegal immigrants.
2. Low-pay illegals also make the warehouses on both coasts (California and Florida) work for the retail industry, especially Walmart (like those low, low pricesÉ)
3. In the midwest, low-pay illegals now do work in factories and meat plants at a fraction of the wages once paid to middle-class Americans.  They do not dare complain about wages and work conditions, for fear of deportation.
4. Millions of American households rely on illegal Mexican household labour.  It`s not just mowing lawns and doing laundry.  Millions of working mothers rely on low cost child-care so they can keep working.  And earning.