Kentucky is arguably one of the health law’s biggest early successes, with about 10 percent of the population getting coverage through the state’s online insurance marketplace — albeit mostly through Medicaid, not private plans — and none of the technology failures that plagued other enrollment websites. The uninsured rate here has fallen to 11.9 percent from 20.4 percent, according to a recent Gallup poll that found only Arkansas had experienced a steeper decline.

But there is little evidence that the expansion of health coverage will help Kentucky Democrats in this fall’s midterm elections. Republicans hold all of the state’s congressional seats except for one, in a district centered in Louisville, and none are considered vulnerable this year. Republicans, who already control the State Senate, have a chance of taking the State House of Representatives, where Democrats hold an eight-seat majority. And several recent polls have put Mr. McConnell ahead of his Democratic opponent, Alison Lundergan Grimes, even though his approval ratings are tepid.

Mr. McConnell and other Republicans here, while more focused on other issues, like protecting Kentucky’s coal industry, continue to attack the health law as a symbol of government overreach and Democratic bungling. And far from flaunting Kentucky’s strong enrollment numbers, Democratic candidates — most notably Ms. Grimes — have remained reticent about the law, even its successes.

In many ways, the role that the law is playing in Kentucky politics reflects what is going on nationally as the midterm elections approach. The law remains deeply unpopular among Republicans and independents, and Republican candidates still use it to flog their Democratic opponents, although not as single-mindedly as before. That is partly because voters are more focused on other issues: A recent George Washington University Battleground Poll found that among likely voters who think the country is on the wrong track, only 5 percent blamed “issues with Obamacare.” More pointed to concerns about the economy, foreign policy, President Obama and Congress.