What should the Republicans do?

About Obama's threatened mass amnesty by executive order?  According to the TV news, Obama will go on TV tonight and announce amnesty by executive order for 5 million illegals already in the country.  He is going to issue green cards, driver's licenses, and everything except health care.  Needless to say this does not meet with approval from the voters, the Congress, state governors, or Republicans. 
  Now that the Republicans control Congress they can retaliate, if they want.  Question:  What can they do to a lame duck president that doesn't anger the electorate? 
  Shutting down the government or refusing to hike the debt ceiling, have been tried in the past, and have damaged the reputation and electoral chances of those who tried it. 
   Strongest power of the Congress is the power of the purse.  If Congress does not appropriate money, the activity dries up and blows away.  There are a lot of things that Obama likes, that are not really essential to operation of the country.  We could do just fine without the EPA, the Federal department of Education, the Highway Trust Fund, the Agriculture department, the SEC, the Davis Bacon "Hire only Union Workers" law, the Dodd Frank "Big bank bailout law", NSA telephone snooping, and the Sarbanes-Oxley "drive mergers and acquisitions to London" law. 
   Congress could simply refuse any funds to these worthless and parasitic operations.  That's veto proof too.
   To exert the power of the purse, Congress must stop passing 1000 page "Fund the entire government" continuing resolutions.  It's too easy to hide money for anything in 1000 pages of government certified gooble-de-gook.  Congress needs to pass separate appropriation bills, one for for each federal activity, that yields a smaller bill which energetic activists have a chance of understanding, and finding the slush funds.  And, each one can be debated and politicked without shutting down the entire government. 
   When the one big continuing resolution comes close to the deadline, Congressmen get asked "Do you wnat to be responsible for shutting down the whole government just to get your petty item on page 754?"  Most Congressmen cave at that point. 
   Whereas if it was just say the EPA appropriations bill, the Congressman could say "Hell yes.  I want this pernicious activity defunded.  If I don't get it, I''ll shut to whole EPA down."