My solution is to force businesses to pay illegals the same as legals. That would illiminate the biggest reason that we can't do anything about illegals. Right now illegals work under minimum wage, so if you need some work done, it costs less to hire an illegal. Since businesses aren't in the habit of overpaying for things they need (including labor), the cut-rate price makes illegals attractive for unskilled labor (things like dishwashers and housekeepers). Remove that incentive, and illegals become less attractive as employees.
Once business no longer have a reason to support an open border, it would probably close at least somewhat.