Would London have allowed you to go to the hospital a second time, or a third time cost free over a number of years? Are you sure your insurance company wasn't charged?
A work permit allows one to work in a country, not live there for life. If you pay into social security and are eligible for benefits by the rules of the country (in the USA, that is paying for 10+ years), you should be able to collect the benefits you have due.
The real question with the anchor babies is what happens to their parents? IMHO, if the parents dragged the kids in illegally, the parents should get a work permit if they want to stay. How racist of me!
Kids brought into the country should also have a work permit, because if they stay, they will someday work in the country. People already here should be able to apply for a work permit in the country without going back to their home countries. Work permits should have a fee that more than pay for their administative costs. Maybe $200 per person / $500 per family which is far less than people pay coyottes today.
Those found without a valid work permit or having overstayed a Visa should be fined heavily - Maybe $2000 per person. If they cannot pay, they will be unable to collect any tax refund checks, social security, or other government payments. Employers found to be employing people without a valid social security number or work permit should be finded $20,000 per occurance.
Problem solved in a human manner --- without a wall.