You're allowed to ramble.
Here I go with another one of those hard truths. I hate it when I have to come up with these, but it happens.
The truth is, not every university student is there because they want to learn. Some are, but some are there because their parents told them to go, or because all their friends were going, or because it's just what you do after high school. These people aren't kids, but I'd hesitate to call them adults at this point. They just don't have the maturity or life experience yet. For most of them, university is their first experience living away from home; it's a lot of firsts, actually. They won't all handle it well. It's not a good thing, but it's how it works.
So, yeah. You obviously can't treat them like they're in middle school, but you can't be too soft with them either. Having a reputation as soft or as someone who fails a lot of students is worse than having a reputation as a hard ass, if you ask me.
Further, if you do get that rep as someone who doesn't (pardon the expression) dick around in class, you'll probably find that the students who do sign up for your class will consist largely of those who want to learn the material. The ones who just want an easy credit will move on to something (or someone) who will give them that.
The part that I think you're still learning is there are times when it's good to be nice and there are other times where it's necessary to be firm and stick to your guns. You need to set clear deadlines and have firm consequences for not meeting them.
There's bullshit at every level, in and out of the education system. All you can do is learn to deal with it, really.
In specific regard to the TA, you've done well so far. But the fact of the matter is, she's pushed it to this point. Whatever the reasoning, she's not getting her work done and that's just not acceptable. You're right; she
has put you in this position. You're not responsible for it. Why, then, should you feel bad for her having to suffer the consequences of her own actions?
On the other hand, if you do end up canning her, how do you think your future TA's will behave? What do you think their reaction will be when they find out that you sacked the first TA ever assigned to you because she wasn't working hard enough?
It's not that you want them to fear you and it's not that you have to be a complete bitch. It's a simple matter of making it clear that if they don't do the work assigned there will be negative consequences. At this point if you don't get rid of her, you may gain a reputation as a 'soft' teacher, one who hesitates to do what's necessary. That's only going to attract more people looking for a free ride. Long term, you're better off getting rid of her and saving you both a lot of aggravation down the road. After all, once she gets out in the world, I can guarantee that any future employers will not hesitate on account of her reputation. If she doesn't do the tasks assigned to her, she will lose her job and that's that. She needs that wake up call and you can give it to her. It would be unkind not to do it, really.