The House does have a role to play in this, too.
Even thought we're talking about the confirmation power (advice and consent) which is only given to the Senate, we are also talking about the Executive "deeming" when the Congress is in a recess so he can exercise his recess appointment power.
Since BOTH chambers of Congress have to concur on calling a recess, the House has a say in determining when it and the Senate are in recess. This is why the House has a play in this power grab.
Problem is what can be done about it other than filing a law suit?
Republican leaders in the House and the Senate can refuse to recognize the legitimacy of these appointments, and publicly declare that they will not accept meetings or debate legislation (perhaps only in the House where they control the agenda) that assumes these appointments to be lawful.
-PJ