My advice is quite simple: run far away from the situation. Anything with that much drama involved is trouble...big Trouble with a capital T.
K and T obviously have major issues. The important point to keep in mind is that T is as much part of the problem as K. She's enabling all of K's drama, which indicates that she gets something from it--and she's likely going to try to get that sort of drama from you, should you get involved deeply.
It's not just an indication that their relationship isn't very good, it's an indication that each of them isn't in shape for a serious relationship.
I also suspect that your partner is struggling with depression. The move from the city to the smaller place appears to have left her reeling. While what you've done may be helping, I think she could probably benefit from a bit of counseling, too, so you may want to explore that with her (and some couples counseling, perhaps).
Keep your stick on the ice. We're all pulling for you.