Wow... you are definitely in a tough position.
I think that K & T have relationship issues that have nothing to do with you and T. K obviously dominates that relationship, and it sounds like she tends to want what she wants but not give any in return. She cheated on T. She has been given a free pass to have sex while she is overseas, but she doesn't plan to extend that courtesy to her gf? That seems just wrong. Did she give T any permission and it's just because it's you, and there are emotions involved? Or was it a flat our refusal - just an "I get to do this, but you don't" situation?
In all honesty, I would suggest that you back away from T a bit, to give her some time to sort out her relationship. I don't believe it sounds all that healthy, but she has to realize that for herself, if it's the case. Regardless, the feelings the two of you have for each other will probably only make it a harder decision for her. It may push her in a direction she otherwise wouldn't go in, and that might lead to resentment between the two of you later. Do T & K have kids together? Because that also would be a challenge, since they'd still need to see each other.
Do you think that you'd have been open to feeling what you do for her if not for your partner's behavior that led to the relationship being opened up? I'm just curious. Is your partner no longer drinking and using drugs? Because that would be a major concern for me given the child involved. It also almost sounds like she was putting you in a hostage situation... she acted out like a teenager and demanded a major change in your relationship - which you granted because it seemed honestly the better choice, between that and the behavior she'd been exhibiting, or just walking away from what had been a strong relationship.
I can see why you and T would be drawn together, given the fact that both of your existing relationships seem to have their troubles. I am glad things with your partner are better, and if you are truly happy in the open relationship, then that is wonderful. It's not that I don't believe it can work, I'm just a little hesitant to say that it IS working, given how it came about and given the intense feelings you have for T now, who is also in a challenging situation with her partner.