I just realized I kind of missed your actual question. Yes, this is absolutely your T's responsibility, and you shouldn't feel bad about pursuing her if you believe that will be fruitful.

But do you have a sense of the probability that she will actually do this? How many months was it that she ignored every single one of your calls and e-mails -- 3 or so? If she did it for 3 months before, how long do you think you'll be waiting now?

I'm a little concerned that you are making this more about contacting T and not quite enough about actually getting the paperwork filled out. But I don't know the situation... perhaps she really is the only person who can do it, or maybe the paperwork isn't actually that pressing or important. If she's the only one who can do it, though, I'd find some way to have a source other than just you put pressure on her. Can you have whomever the paperwork goes to contact her? Does she have a supervisor?