The biggest obstacle I see here is the fact that she's being sent to a world where politics and wealth are a big deal and she has no knowledge of that world. She wouldn't know who to align with and what to watch out for, and getting all of that knowledge without meta resources would be tough. Making an alliance with someone without having the same knowledge most of us do is risky, she may be cunning and clever, but so are many of the players in Westeros.

She could easily take anyone in Westeros in a fight, but I can't see her taking down large scale armies without getting hit at some point and then overwhelmed. She could use her bending to gain influence, but I feel like most of the houses who would align with her would see her as too big a threat in both fighting and intelligence to let live, and have her assassinated in some way. She's a great fighter, but you can't fight off something you never see coming.

It's possible she could turn the tables on someone who's trying to use/fool her, but it just seems too unreliable when they'd have more resources. Even R'hllor's worshipers are a bit iffy, they may think she's an emissary or something, or she could be outed as a heretic.

Basically, I'm not sure she really can. I think she's clever enough but doesn't have knowledge of the world or politics to match the other clever contenders. She's the deadliest fighter by a mile, but doesn't have then sheer resources to stomp out the other fighting forces. Combining her advantages could help, but could backfire once the others are aware of her.

If she's patient, maybe she could get somewhere in a year, with most of that time spent sending Mai and Ty Lee off to gather information about the world and the people, and to steal valuables to build up some funds.