Hey, I've got an idea! Let's give Michael Flynn immunity! Then he'll spill the beans on Trump to Congress and Trump will get impeached!
No. Anyone thinking that has been watching too many courtroom dramas. (Whatever happened to Fred Thompson's presidential campaign? Oh, yeah...)
Have you ever watched a guilty person testify before Congress? They just plead the Fifth Amendment, over and over and over again. It may look bad, but it doesn't matter. And it doesn't matter to Flynn. He has already resigned in disgrace, you can't convict based on a Fifth Amendment invocation-- which is the whole point-- and there really isn't anything beyond that. So what's Flynn's game here? Two possibilities.
1) Flynn just wants to guarantee that he is never prosecuted under the Logan Act. We already know that he violated the Logan Act. Right after Obama imposed some mild penalties for Russia hacking the DNC, Flynn called the Russians to tell them not to respond because Trump would undo everything. That was a private citizen engaged in diplomacy to undercut US foreign policy, which violated the Logan Act. In principle, he could be prosecuted, but not if Congress gives him immunity, and he then tells Congress that he did it, on his own, with no instruction from Trump. That way, he gets off scot-free, Trump stays in the clear, and everybody in Trumpland is happy.
2) There is something beyond the Logan Act for which Flynn wants immunity.
Possibility 1 is the most likely. The chances of a prosecution under the Logan Act were never high. It is not exactly a high priority for federal prosecution, for a wide variety of reasons that others can explain better than I can, but Flynn is probably fine now. Still, even if you think there is only a 1% chance of prosecution, why not reduce that to 0%? Would you trust Trump to pardon you? I wouldn't trust that sack of shit to do anything for me. So, try to weasel an immunity deal from Congress with the intention of giving them nothing of value anyway.
What about Possibility 2, though? Does Flynn have dirt on Trump? Of course he does. Would he share that dirt in exchange for immunity? No. Not now, anyway. The immunity deal is a play that someone takes when his back is up against the wall, and it is that or prison. He doesn't flip on Trump unless he is otherwise screwed, and the only thing we know about Flynn is the Logan Act violation. And again, he was never likely to be prosecuted for that, so that isn't enough to get him to turn Henry Hill.
If Flynn were facing real prosecution for something, an immunity deal might make sense, but he's not. He violated the Logan Act, and that wasn't likely to be prosecuted. So, he has nothing for which he needs immunity. If he were brought before Congress without immunity, he'd just plead the Fifth. Most likely, he just wants to reduce the probability of a Logan Act prosecution from 1% to 0%.
Oh, and Oliver North kept his mouth shut and did his time without flipping on anyone higher, so even if Flynn did get caught for something beyond the Logan Act, he still might not flip.