You've already gotten some great advice here, so I just want to add a couple of things.
Good egg fertility since last mating in Mallard-derived ducks like yours ranges from a week to 10 days, in general. I've read things on the Internet about it lasting for up to 2 weeks, but I think that's the exception. As far as your duck not being broody, Pekins are not great for going broody, as @Pyxis
mentioned, but even ducks that tend toward broodiness don't set on just one egg. They lay enough to have a clutch they're satisfied with before they set on them. (I have no idea how they decide how many is enough, and it can vary dramatically with the individual duck.)
As big a fan as I am of the high-end Brinsea incubators, I wouldn't recommend you get one of those right now unless you're sure you'll use it in the future for a lot of incubating. You'd be looking at spending $300-400 USD for an incubator that might not hatch anything this time. I say that only because I want you to try this but don't want you to be devastated if nothing comes of it. The first eggs of the laying season often won't produce viable ducklings, and that's even more the case with the very first eggs a young duck lays.
Any chance anyone you know, or maybe even a local school or co-op, has an incubator you can borrow or rent inexpensively? That would give you a chance to try it with your little girl's eggs without sinking a lot of money into it and feeling like you wasted it if it doesn't work out this time.