Just as many have written, generally speaking, there is no obstacle to solving all this.
Go after the employers. If illegals are unable to work, they will self-deport. Before the welfare state emerged, this happened each time the U.S. went into an economic plunge.
To minimize the welfare state's safety net, require applications for assistance to be processed through INS. Tell hospitals and clinics, yes, they are responsible for proof of citizenship. Tell them it is a form of health insurance.
The unwillingness of Americans, or at least their representatives (even in the House), to accept the measures necessary to meaningfully penalize employers suggest to me that they are not willing to sacrifice all that much, in terms of liberty and inconvenience, to curb illegal immigration.