Like others have said, let your daughter decide, though I would also add the "Ask your wife if she wants to be seen in prison." It might not be something that she wants her daughter to see, so I think it would be polite to run it by her casually. If I had to guess, she'll probably want to see her daughter very much, but it could be something she's ashamed of, and wouldn't want her daughter to experience.
But yeah, if all parties are on board for it, let her do it.
As far as preparation for the visit, I can't speak for prison, but I can speak for mental institutions for the criminally insane, which are pretty damn close to prison. I had to visit my brother in those type facilities for the past few decades.
Tips to make the trip much easier.
1. Don't bother bringing your purse, phone, keys, etc. Just don't even bring them in the facility. Leave them in the car, because they're going to likely take them off you anyway while you are visiting. You can maybe bring in some change, to buy snacks from vending machines. This may or may not be available, depending on the security level of this facility, and how "family friendly" their visiting facility is.
2. The place will feel very confined (duh), but it will also likely smell strange. I don't know about prisons, but mental facilities have an...odor to them, from the large number of inmates, who likely don't have good hygene, it will create a pervasive scent to the place. I'm not saying it's going to smell like piles of rancid feces or something, but it might have a smell that just sort of hits your nose wrong, and slightly put you off things.
3. Her mother is likely going to be very nervous about being seen in that situation, at least at first. So give her some slack when it comes to awkward behavior until she settles down about seeing family while in prison. Alternatively, you or your daughter might feel like this too. The corrective system is an intimidating environment until you get used to it. Just try to relax, and understand that there will be people with weapons around, watching you and the inmates at all times, or at least while you transit through the place.
4. Call ahead to the facility and ask if there are any things that you are permitted to bring during your visit. Perhaps a book, or maybe some homemade cooking for her to eat while you visit. Little things like that can help to make the visit feel more natural, as you discuss the gift, and also provide a break from routine for the mother, cause trust me, that food isn't all that good. If you can bring food, you might try having you and your daughter cook it together, so your daughter can maybe tell her mom about the ordeal of trying to cook the stuff for the first time. I'm assuming your daughter isn't a skilled cook of course, which if she's not, it might make for a humorous story to share with mom while she eats it.
5. Basically, just try to relax and not let the place put you or your daughter off too much. It's never going to be a 100% fun place to visit with a loved one, and there will always be some awkwardness to it. But if the 2 of you are genuine in your attempt to try and make mom feel better, the visit shouldn't be too bad. Considering your daughter was able to make a joke about her mom now having to follow rules, instead of handing them out, it sounds like she's pretty well balanced to the situation. I would maybe remind her that there are millions of people in this country who are criminals, and that doesn't make them "bad" people. It just makes them people who broke a law. And that has consequences. I'm sure she's disobeyed some rules in her life by 14, so I'm confident she can understand this, and that her mom has to pay for her crime by being stuck in that place for a long period of time. Basically an extended "grounding" if you will.
Hope that helps.
