A resident tries to resume an application he started on the Maryland Health Connection insurance exchange Web site at an education and enrollment event in Silver Spring, Md., on Dec. 7, 2013. (Andrew Harrer/Bloomberg)

Members of Maryland’s House of Delegates on Wednesday said they plan to forge ahead with questioning the leaders of the state’s troubled health insurance exchange, even though days earlier a high-ranking state senator said the General Assembly was largely done with its inquiries.

Last week’s announcement that lawmakers would defer most questions to a regularly scheduled state audit, which would probably start in the summer and could take a year, sparked anger from Republicans and some Democrats who want a full accounting now. Some accused lawmakers of trying to protect Lt. Gov. Anthony G. Brown (D), who was tasked with implementing President Obama’s Affordable Care Act in Maryland, as he runs for governor.

On Wednesday, a group of House Republicans also filed legislation that calls for a joint committee of senators and delegates to produce a report on the exchange’s problems by the end of the year, said House Minority Whip Kathy Szeliga (R-Baltimore County). The committee would have subpoena power and could “compel the attendance of witnesses,” she said.

“It’s not a partisan issue,” Szeliga said. “This is the biggest single program of this administration. It’s a lot of money, a lot of time and deeply affected thousands of Maryland residents.”

House Speaker Michael E. Busch (D-Anne Arundel) said delegates have been working for weeks on a plan to identify “what things were problematic and how to avoid them in the future” but are not ready to announce what form that will take.

Del. Peter A. Hammen (D-Baltimore), who is in charge of formulating that plan, said he still has a list of questions for health officials, including how much the state has spent on the exchange, which contractors are still working for the state and what technology problems remain. Hammen said these questions would probably be best handled in a series of briefings attended by senators and delegates and likely starting next week.

“That was what I always wanted to do,” said Hammen, chairman of the House of Delegates Health and Government Operations Committee, one of two committees that have overseen recent legislation related to the exchange.

Last week, Sen. Thomas M. Middleton (D-Charles) said he didn’t think that multiple hearings would be necessary. Middleton, chairman of the Senate Finance Committee, said that there was no need for a special investigation and that the routine audit would be sufficient to look into the exchange’s problems. Dwelling on the topic, he said, would likely only serve as political theater between candidates for governor.

“I’m not looking to make excuses for it,” Middleton said of the exchange Friday. “But I don’t see anything that warrants a full-scale investigation.”

Brown’s Democratic rivals for governor — Del. Heather R. Mizeur and Attorney General Douglas F. Gansler, both of Montgomery County — have called for an immediate and deep review of the exchange’s ongoing problems.

Gansler’s running mate, Del. Jolene Ivey (D-Prince George’s) said that earlier briefings on the exchange seemed to be for show and that she’s looking forward to a fuller explanation of the exchange’s problems and if they can be fixed.

“It’s really overdue, and it’s ridiculous that it has dragged out this long,” Ivey said. “It seems like a lot of interest in saving face, and I just don’t have any patience for that.”

Maryland was one of 14 states to launch its own health insurance exchange, created as part of the Affordable Care Act.

Although Maryland leaders had promised that their exchange would be one of the best, the Web site crashed on its first day, and it has been difficult for many Marylanders to sign up for coverage. Even after months of improvements, enrollment numbers continue to lag behind expectations.

As of Jan. 25, officials say, 26,832 Marylanders had enrolled in private plans through the exchange. Since the new year began, enrollment has slowed, and the state is not expected to hit its original enrollment goals of 150,000 to 180,000 by March 31, when the first enrollment period ends.

The second enrollment period will begin Nov. 15. Starting in April, state officials will have to decide between continuing to fix the exchange — which its chief information officer says has “a number of architectural and infrastructure issues” — or trying something different.

There have been calls for Maryland to switch all or part of its system to the federal exchange or to evaluate joining another state’s exchange.