My answer to this depends on what she means by "competitive gymnast."
If her goal is to be a serious upper-level athlete (which is very unlikely, but not impossible, for an athlete getting into competition as late as 13), then she should not train roundoff back tuck. It reinforces a lot of habits that will cause problems in the development of powerful high-level tumbling. As a coach, any time I see a kid with high-level aspirations do roundoff-tuck, my heart sinks, because I know I'll have an uphill battle in teaching them to generate power in a roundoff-backhandspring.
For a 13-year-old who is still new-ish to the sport, the most likely competitive route (I'm assuming we're in the USA here; if not, none of this necessarily applies) is an Xcel, which is a more-or-less casual competitive program. And for a kid who doesn't necessarily need to prepare for eventual big high-level tumbling, roundoff-tucks are just fine. If this is the case, then here are my suggestions:
First, I am not a fan of self-taught back yard tumbling in general. A lot of tumbling technique is counterintuitive, so she's very likely to teach herself incorrect technique. And it's harder to correct bad technique than it is to train good technique from scratch. So the best route forward, if at all feasible, is to not train the skills at home, and see about getting into the gym more often.
However, she may the sort of kid who's going to do it anyway (and I must admit I was exactly this same sort of kid when I was that age). So while I think training it at home is a bad idea and I don't encourage it, if she's going to do it anyway she might as well do it properly. Here's how I'd go about it:
1) Let's separate out the back tuck from the roundoff and make sure it has good technique. Standing back tuck works for this, but doing it on a trampoline works better. The biggest thing to focus on is takeoff position. Arms up, head neutral. The most common mistake in self-taught tumbling is to lead with the head. She should NOT be throwing her head back as she takes off for a back tuck; rather, she should be watching something in front of her for as long as possible, so her head is the last thing to start flipping. She should practice this until it feels intuitive and easy and not-even-remotely scary.
2) Heavy emphasis on use of visual cues. She should see something in front of her as she takes off, she should see her knees going over top, and she should see the landing before she lands. There should be no guesswork when in the air.
3) Ab strength. Hollow rockers, crunches, V-ups. These will help her to get her knees in quicker.
And let me emphasize one more time: I think training this skill at home is a bad idea from both a safety perspective and a developmental perspective, and if she has high-level aspirations, it would be better not to train roundoff back tuck at all, and focus instead on backhandsprings.