In some ways, it surprises me that they didn't just walk away. It's not exactly their crisis since they put the blame on Obama's unilateral (executive) decision to end deportations of illegal immigrant minors. The GOP could say "you broke it, you fix it" to Obama.
They haven't done it because of a little lawsuit they've started which complains that the president is overreaching his authority beyond what is legal. It's hard to make that argument stick when Congress won't address issues and pass laws, which inevitably leaves the president to sort through the mess as well as he can. This will entail making unilateral decisions, seeing as the Congress hadn't passed the laws he'd need as the guide. So it's important for appearances for the House to pass something.
There's no certainty that the House can pass any sort of emergency immigration bill. They can afford only about a dozen defectors, and there are at least that many angry Tea Partyers or assorted nutjobs. So it's going to be tight for the GOP leadership. I don't think they're going to succeed, even though it hands Democrats a talking point about the disorganized House that can't provide leadership even in a crisis.
A leopard can't change its spots, and the House can't pass anything controversial unless absolutely necessary. If it's not a necessary budget bill, it's not going to happen. Maybe I should start my August vacation by chortling at Congress while I lounge poolside with a martini. Yes, that's the ticket.
The House sorts it out and goes on vacation... soon... maybe...
Update 8/1/14. I was wrrrooonnggg! The House GOP got it together and passed a bill. If the House can become functional, the Dems better watch out in 2016.