Granting the illegal aliens and undocumented workers papers to stay is rewarding them for breaking our immigration law. Keeping them for cheap labor is false economy because they caused an enormous economic burden to the tax payers. Once they have papers to work and stay, they will demand the minimum wage or higher, and demand for any form of welfare services they can get.
I don't believe it is possible to find hard evidence that proves that undocumented workers cause 'an enormous economic burden' to the US tax payer. Due to the emotional nature of the issue, and because there exist many unknown variables (the exact number of undocumented people in the US for example), burden/benefit estimates necessarily incorporate subjective presumptions. For example, I believe that the work that these workers do is in itself an enormous contribution to the American economy. In fact, several large American industries would collapse without these workers.
It is obvious that undocumented workers do not or cannot avail themselves of the substantial majority of benefits available for Americans. Furthermore, the costs of services that they may happen to utilize (school services for their children - who may well be US Citizens, ER rooms for traumatic injuries, etc) are exponentially outweighed both in numbers and proportion by US Citizens on welfare or other form of public assistance. Undocumented workers pay for their own food, clothing, rent, vehicles, gas, routine medical checkups and other services without the benefits of public assistance. Said money directly and indirectly benefits a large number of individuals/companies, most particularly the provider of the goods or services and the US economy in general (for example, the provider of the goods or services pays taxes on the income, jobs are created and maintained).
Finally, it is apparent that undocumented workers would be happy to pay more than their fair share of taxes/fees; even social security (to which they will likely never have access), if they were able to do so without fear. I am of the belief that the vast majority of undocumented individuals are motivated by a burning desire to create a better life for themselves and their family through honest and hard work - therefore I do not believe that they would, as you mentioned "demand...any form of welfare services that they can get". There is no reason why they should not excel in the US if given an opportunity, perhaps in the form of a guest-worker status. The US economy and tax base would also stand to gain in the tens of billions every year. This fund could be used to help implement proper immigration policy/border policing & supplement Medicare and social security.