Will drum brakes on replacement Sturmey Archer 5-speed hub improve with use?

Yes, that is true. Those types of brakes require a "burn in" period of around 50 full-speed stops.

I never had a problem with the brakes on my SA five speed, but I could never get it adjusted properly to go into first gear. I don't think the five gear series being sold now is one of SA's best products. If it's making a squealing noise there may be water in the hub or a lack of grease. If the problem is a lack of breaking power that may be the nature of the beast. Drum brakes won't stop as well as other types. If it doesn't work at all, then it is a defect or improperly installed. Go to bike forum.net and do a search to see if someone is having similar problems. If yours is unique, start a thread and ask your question there. A lot of good mechanics, pro and amateur, help out there.

I've heard lots of people mention the more than lacking braking power with Sturmey's. I've also heard that braking power improves (or is noticeably more affective) the faster you go with these types of brakes.

What exactly is the brake doing or not doing?

Yes they will improve with use. Have her go out and stop and start quite a few times. The same thing happens with disk brakes but for a different reason.

You should never rely on rear brakes alone to stop, you should also use the front brake which has at least twice as much stopping power as the rear brake. the rear brake will probably improve as the brake shoes bed in but your daughter should also be using the front brake whenever she needs to stop quickly

drum brakes are never very good.

wle