I'm worried that I've lost my chance to finish the Markarth city quest line.

When the Markarth town guards tried to frame me at the end of the "Forsworn Conspiracy" quest, I fought back. Then, rather than fight my way out past every guard in the city and probably slaughter a few civilians as well, I took an invisibility potion and snuck out the city gate, where I had to kill a few more guards before I was able to get away.

Now if I return to Markarth, I am given the option of surrendering and going to prison when I first encounter a guard. However, when I do this, the quest "No One Escapes Cidhna Mine" does not begin.

I am taken to Cidhna Mine like a normal criminal, and a few elements of the quest are present. A guy tells me how to get some skooma and a shiv -- which I know is how I'm supposed to get past the orc to meet Madanach, the Forsworn King.

However, no one mentions Madanach in any way, and once I have the shiv, I am not given any opportunity to use it to get past the orc.

Have I closed off this questline ever by leaving the city after the first stage of it?

Or is there something I'm just not doing once inside the prison to trigger the quest?

You can always load a previous save game. Have you tried doing that yet?

I think you closed it off by not accepting arrest right away.

Bummer. It's far too long ago now to go back to a previous save. I guess I'll have to use to console to force a quest stage.

This seems to be a glitch since my Nord Warrior massacred the Markarth City Guard when they tried to arrest me and he subsequently waltzed out of Markarth leaving a trail of dead guards behind him. However when I returned I realised that I had a 9000 gold bounty on me and again I fought my way out again rather than go to jail but in the end I had to go to prison since I needed to complete quests in Markarth and the quest started normally for me.

Unless something's changed in one of the recent patches, I don't think this is the case... on my first playthrough I did exactly what he did, fought my way out of Markarth after being ambushed by the guards in the Temple. Went back there on another quest later, guards went aggro, I submitted and got arrested and sent into the mine and the quest started and finished fine.