N.B. The links to terms are for third parties, rather than the questioner, who as 7 kyu will already know them.
Where she is now
The best approach depends on the sort of person she is and what she hopes to get from playing go, so stay alert to any feelings, wishes and opinions she expresses. As Forget I was ever here’s answer says, I think you should find out what she thinks about your ideas by asking her as well as by staying alert.
It sounds a bit as though she is not too bothered (yet) about improving, given that she plays fast, repeats mistakes and does not mind losing. Perhaps at the moment she just enjoys the feeling of placing the stones, the patterns they make, capturing stones and maybe some simple problems. If that is the case, you should be patient; for now, I would keep the games fun and let her see she can learn more when she wants to.
Try not to be too disappointed if she does lose interest in what she enjoys now but does not get interested in improving. It would be great to have a go-playing neighbour, but it all depends how she develops personally.
Learning from mistakes
In friendly games (which they probably all are), when she makes a mistake for about the third time I would ask/tell her something like “are you sure you want to do that?”, “last time you did that it turned out badly!”, “do you remember what happened last time you did that?” or “can you see why that doesn’t work?” without saying what the mistake was. If she does not like that, you could try just saying “Aha!”.
As long as she just plays for the fun of it without really wanting to improve, I would be fairly easy-going about letting her take back moves, especially if you have pointed out the mistake. But if she starts showing off (too much) about how well plays against you, you could suggest a match on stricter terms, i.e. still with a handicap, but no taking back moves once she lets go of the stone, but probably not a clock unless she shows interest in trying it out.
Given her current attitude, I would be surprised if she wanted to review an entire game. What you could try is to analyse the last mistake she makes, especially if it causes her to lost the game. Perhaps “Do you want to see how you could have won if you had played your (e.g. third last) move differently?”
I would not expect her to be very interested in joseki, but you might be able to work in a little tsumego if you do review a game, by asking her to find a better move in a position from the game, and if that interests here, changing the position and see if she can spot a geta or a snap-back.
If you want to move her on from capturing, I wonder if it might help to show her a throw-in; it is still about capturing, but letting your opponent capture first shows that capturing is not always a good move. A next step, if you can manage it, might be to engineer a situation at the end of a close game where she can win by making a big move instead of capturing.
You may want to check out the Teaching Methods page in Sensei’s Library, though the section on teaching children does not seem to provide much specific help.