Congress should exercise its power of the purse. Send them another written demand, with a do-by date. When they fail to meet that suspense, slash their budget by 10%. Follow up immediately with another demand letter, with an even shorter suspense date. When they again fail to meet that, slash their budget by another 10%. Lather. rinse, repeat.
At some point, to meet their lower budget authority, they'll have to start laying off people. At that point, lower-level minions, seeking to keep their jobs, will suddenly start "leaking" the requested material out of a sense of survival.