Are you sure they're both smartphones?
Supposedly AT&T won't activate a smartphone on its network unless you add a data plan. As far as I know, that's been their policy since at least 2008. Maybe your wife's phone just happened to fall through the cracks? If I had to guess, I'd say plugging your wife's SIM into your daughter's phone won't be enough to get things working. You may even wind up with an additional data plan added to your account. Only one way to find out. Let us know how it goes.