I recently started playing in Nightmare mode for the first time, and have been playing through the entire Core/Mirkwood cycle straight solo (i.e., one-handed), progression style (including only cards available at the time of the scenario's release).
I knew that even more than its standard version, Escape From Dol Guldur NM was going to be a beast to get through. I almost skipped it so as not to discourage myself. But I had beaten the original solo with Core set cards only, and after defeating both Journey Along the Anduin NM and Conflict at the Carrock NM, I thought, "oh what the hell, why not"?
I played the quest with a strict interpretation of the rules as written (prisoner was always randomly drawn, I played on standard NM difficulty, I played no more than one ally per turn, no house rules to make things easier, etc.). I experimented with a number of decks, and played the quest 45 times before I scored a win. This particular deck scored a win on its 18th attempt (I always did better when Glorfindel was the captive, and the one victory was no exception). So no, this deck does not reliably beat the quest solo, but what deck does?
The main strategy? Aside from Cavern Guardians and maybe the occasional Torture Master, there is no point in trying to kill enemies (your allies won't be up to the task), and getting rid of Crazed Captives can do more harm than good (yes, they are a delicate balancing act) . Keep as many enemies at bay as you can with Frodo loaded up with Unexpected Courage x3 and hopefully buffed by Arwen Undómiel. Quest as fast as you can and get the hell out of there. If possible, claim Gandalf's Map first, with Éowyn. Frodo Baggins is your best bet for claiming the Shadow Key. It doesn't really matter who gets the Dungeon Torch... like the Shadow Key, that one is best claimed when you're ready to break out of stage 2 and straight to freedom on stage 3 (assuming no Dungeon Jailors forced you to claim the objectives sooner). Play Saruman (beware his Doomed 3) when you are ready to make a break for freedom and victory on Stage 3, so that he can place the Nazgul of Dol Guldur (who is NOT a unique enemy!) out of play.
I did "cheat" somewhat with regards to my own self-imposed restriction regarding the "progression" requirement. After previous attempts where I consistently failed to get a good sequence of Song of Kings, Sneak Attack, and Gandalf with which to defeat the Nazgul, I subbed in Saruman from The Voice of Isengard instead... which was technically released 6 months after EfDG NM. It's the first time I've ever had to do that, but oh well... I saw no other viable solution.