Between chapter 80 and 81, here's my analysis. I can think of seven broad possibilities;

1.) Do nothing
2.) Attack publicly
2b.) Attack publicly in disguise
3.) Stealth attack
4.) Retreat and regroup
5.) Change the board
6.) Deus Ex Machina

1.) Do nothing; I list this simply because people often forget that inaction may be the best possible action. Here, that doesn't seem to be the case. On the other hand, once you realize that sacrifice is necessary, why not give in to the dark side? What's one muggleborn more or less? With proper obliviation Harry can literally forget about Hermione. Plus, the dark side has tasty Hufflepuffs. And cookies.

2.) Attack publicly.
While romantic, this puts Harry into a massive, wasteful, battle with basically all of wizarding Britton.
He's good, but realistically, he'd lose.
Blackmailing the council publicly seems equally pointless. Even if they gave in, it would be disastrous in the long run. On the plus side, this is by far the most dramatic possibility. It's not hard to imagine Harry laying waste to the Dementors essentially freeing all the prisoners, and throwing the wizarding world into complete and utter chaos.

2b.) Attack publicly in disguise. Basically, put on a mask and break Hermione out of custody. Again, several possible ways to do it, but all with the significant drawback of making Hermione a wanted fugitive.

3.) Stealth attack. Harry and Quirrell almost succeeded in getting Bellatrix out without any help and without anyone knowing. With the order and the aurors attempting it, it wouldn't be unimaginable that they could remove Hermione without anyone finding out. On the minus side, Hermione would have to become a non-entity for 10 years, and they'd have to sneak her back in. On the plus side, the comedic potential is enormous. Almost every major character could reasonably have a motive to sneak Hermione out, even the evil ones. Massive Gambit Pileup ensues.

4.) Retreat and regroup The bad part of this is Hemione will be in Azkaban for some amount of time. The good part is that it doesn't result in Harry being at war with magical Britton before he's ready. This probably isn't as bad as it first seems. Many will be outraged by the decision to send a 12 year old girl to Azkaban, including, I bet, Draco Malfoy. Now that the court room acting is over, "Draco" can argue for leniency, and Lucious can soften his heart over the plight of so young a girl who was clearly unaware of the severity of her crime, and blah blah blah. Lucious wants Hermione away from his son, discredited, and wants to stir up the blood purists. All that is accomplished if Hermione is locked up in, say, Nurmengard, and his son vows to stay away from her. Add in a bad publicity campaign smearing the wizingamot "Malfoy says 12 year old girls should be tortured in Azkaban". Harry might even be allowed to visit and banish a few dementors, rather than having to do it clandestinely

5.) Change the board. Determine who really cast the blood chilling charm. Find out who killed Narcissa Malfoy, and give them up. Or the dark path - find someone and give them up as Dracro's assailant/Narcissa's killer, without considering their actual guilt. (Harry could put his minion Lesath Lestrange to good use.) Find something else that Malfoy wants, like say, the philosopher's stone, and give it to him. Become god. All good things to work on, but their timing is not under control, which means this really is a variation of one of the other options with extra work added.

6.) Deus Ex Machina. The author could make anything happen. While it might be that only the author can save them now, it's not something I'd expect the characters to plan for. And I for one would feel cheated if that was the final solution.