I'm going to assume that Artemis is simplifying the situation by putting it on a level Ryza would more easily understand - that Mantrae has betrayed us to another state (my bet remains as it has been from the start it's the Empire) and the bandits are simply in their employ. Which is to say, the left-field option of Ryza going "Oh, them? One sec." heading the direction they arrived from to run back into them and go dragon on them probably doesn't solve the situation.
...Unless we want to gamble that wiping them out is read by their employers (who probably won't know what caused their demise) as the Countess reneging on their arrangement and forcing her to flip back to our side out of desperation. But other than the fact I just like the concept of trying to solve the problem with something more out-of-the-box that seems a poor gamble to me, far too many points of failure and reliant on people we don't know well behaving the way we want them to.
So, uh, options within the castle and its nearby environment. It's extremely cold and I'm not sure it would be in either Ryza or Artemis's character to do so but breaking Artemis out by force, up to and including just going dragon and busting a hole in the wall, and bringing her to her father might not be the worst move here. We'd be leaving the others behind temporarily but if Artemis is correct there's really no reason for any harm to befall them other than spite. It's Artemis they want, turning over the others isn't going to get the Countess her daughter back. I think the major risk here would be them demanding them anyway for retribution for our previous encounters with them. And we'd be coming back hopefully with an army at our side.
Getting everyone out complicates things a lot, I'm not even sure where we'd want to begin. I'm assuming it's out of the question we can directly break into Artemis's room from the outside without raising an alert. So from there I think the most promising way to get there would be to hope there's an unlocked entrance at the top of this tower, or a guard with keys, enter from there and hope we can get to Artemis quickly from there. We'd undoubtedly face some guards outside her room but it's not impossible we could ambush our way to this point. But once we've got her we're stuck at "where's everyone else?" with the added problem of "Artemis can't fly" limiting our mobility. The idea that we could stealth our way through the interior seems... unlikely, if anything I'd expect the density of guards to increase as we descend. We could ask Artemis if she has any idea where the others are being held but she may not know and even if she did, decent chance it's like... the keep dungeons... and Artemis is just getting the high value/noble prisoner treatment with this tower. And on top of that we have circumstantial evidence to believe Kelton isn't with them.
All this to say that unless we think we can just RAWR DRAGON our way through this castle I'm not sure I like our odds of finding everyone and getting out. I think we could probably get Artemis out and take her with us, but beyond that I don't have any good ideas.
Question though because my closest thing to an idea hinges on it: Does Ryza believe she could force her way into this room (and not collapse the tower) if she went dragon form and tried to break in? Because if she can I think it's plausible we could bust our way in and fly off with Artemis without blowing our cover. The evidence will tell them that somehow we were responsible but I think Occam's Razor would tell them we're some crazy artificial mage like Artemis originally thought rather than dragon.