Congress has the constitutional right to declare war. If the GOP "conservatives" were really serious about fighting illegal immigration, they would have the moral courage to openly state that such "immigration" is tantamount to an invasion, analogous to a military invasion, and they should vote in Congress to declare war on the illegals (can Congress declare war on extra-territorial entities?). If the President would then refuse to order the US military to enforce that declaration, then that would be reasonable grounds for impeachment. In other words, send the US military to the border. If the invaders resist, shoot them, if they surrender, put them in POW camps (under minimal conditions) until the war is won and they are repatriated.
The other possibility would be to declare war on, say, Mexico, but that's getting into a more gray area of international law. Is Mexico aiding and abetting the illegals' invasion of the USA? Can it be proven to the extent that would justify a military invasion of that country?
The more direct approach - that Congress has the right to declare war on extra-territorial entities and proceeds to do so - would be best.