The February study by the Institute on Taxation and Economic Policy says that, on average, the nation’s estimated 11 million undocumented residents pay a total of about
8 percent of their incomes in state and local taxes each year[0].
That's total taxation, encompassing sales tax, and (for those not being paid "under the table"), income tax and FICA. Again, that's a national average; in CA, sales tax would make it higher). That's not remotely paying all of their own way overall.
As for overall impact: "Ira Mehlman of the Federation for American Immigration Reform said the $11.74 billion tax contribution covers only a small fraction of what undocumented residents and their dependents use in government services. The Washington, D.C. organization, which favors stricter immigration enforcement, estimates it collectively costs state and local governments about $84 billion a year to provide education, health care, public safety and other services to undocumented residents and their dependents."[0]
As for Social Security, they get that back as much as legal residents do, if they file for it. Quoting your source: "He calculates that undocumented immigrants paid $13 billion into the retirement trust fund that year, and only got about $1 billion in benefits." That's retirement benefits alone, being paid out already. As these payees age and retire, they'll collect far more than they paid in, and that will worsen as lifespans increase... hence it being referred to as a "Ponzi Scheme".
"We lose money on every transaction, but we make it up in volume!"
[0] SJ Mercury News, March 3, page B2, "Undocumented immigrants’ taxes calculated"