Is she directly under you? Do you decide on her salary and review, and whether she continues to be employed by the company? Never mind if it's with the help of someone or if you provide input but aren't the final decision maker. If you are anywhere in the process, then you're not being a whiny bitch to ask for people to do what you need for them to do, and not go off on their own little tangents.
Is there a process for working for clients? Are proposals written down, stored, etc.? Are they vetted before going to the client? Then this employee has likely ignored the process. And this is what happens when you do.
If there is no such process, then use this situation as a learning opportunity and create one. I know that doesn't help you now, but it'll help in the future.
Now back to today.
Have a meeting with this woman. Less about her tossing her 2 cents into your training documents. More about the chain of command/sequence of operations for doing things. Even if there is no procedure in place (yet), going rogue isn't helping anyone. And overpromising and underdelivering is the polar opposite of what you want to do. Work out with her a plan about how the two of you, together, are going to fix this clusterfuck. Give her a short-ish deadline to get things done from her end. Not so short that it's impossible for her to meet expectations. I mean more for her to feel a bit of the pain of this problem, perhaps staying late, etc.
Note: I'm not suggesting being vindictive. It's more that if she skips away with no pain or consequences, with lots of time to do very little (or nothing), then she won't learn from this experience. And the short-ish time frame is also for the benefit of your client. Your department messed up. The client deserves as quick a fix as you can give them, regardless of how or why or who caused the problem.
Finally, don't expect congratulations or the like from people you've been working with if you were promoted from the pool and they were not. It's possible they were also vying for this role. They may be resentful. Don't stoop to that level.
At the same time, though, keep in mind that you're not pals any more, if you ever were. You are now boss and employees. This means you call the shots, you ask them to do stuff, and they do it. You can be cordial (you should be). And you can get input from them, and sometimes accept it over your own ideas (another thing you should do). But the power dynamic should be obvious. Right now, it isn't.
Being a manager doesn't mean choosing to be nice but a doormat, or mean and hated but at least listened to. You can straddle the middle line. Be assertive. Make your requests crystal clear. Praise people for jobs well-done, and do so in public. Confer with them about mess-ups, but do so in private.
Be. The. Boss.