Does the wire cage have to be pretty big?
No, just big enough for her, food, and water. Nothing that looks like a nest.
How long does broodiness last?
That is a hard one. The short answer, could be a few days, could be a few months, with a month and a half probably more normal.
Before a hen even starts to lay she builds up a reserve of fat. Even when they are not laying they have more fat than a rooster, I've butchered enough of both to know. A lot of that extra fat is in a "fat pad" in the pelvic area but more is spread through her body. That fat is what the hen mostly lives on while she is broody. That way she can spend her time on the nest instead of going out to look for food or even water. Broody hens will lose weight while broody but that's just fat put there for that purpose. It's not a threat to her health. She will have to replace any fat used while broody before she starts to lay again so the sooner you break her the sooner she gets back to laying.
Broody hens should leave the nest to eat, drink, and take a dump. A broody instinctively knows to not poop in her nest but holds it. I've had broody hens come off the nest once a day for about 15 minutes, especially in cold weather. In the heat or summer I've seen a broody regularly come off twice a day for over an hour each time. Some broody hens I never see off the nest but since they are not pooping in the nest they are coming off. It doesn't hurt her to toss her off the nest a time or three during the day but you don't have to. Broody hens have been doing this for thousands of years and instinctively know the process better than we do.
Most hens will break from being broody when that excess fat pretty much gets used up. The more they eat while broody the longer being broody can last. Some will break on their own before that, some will go even longer which can risk their health. How long does that fat last? Some hens store a lot more fat than others. Some hens eat more while broody than others. So it varies. I'm pretty comfortable with a hen staying broody for 5 to 6 weeks. If they haven't hatched by then I break them.
The other part of this is that the hormones that cause them to go broody affect different hens differently. Some slowly work their way into becoming broody. They may show some signs of being broody for as long as two weeks but never flip over to full broody mode though sometimes they do eventually flip. These are typically pretty easy to break, sometimes just by tossing them out of the nest a few times before they get set. With some the hormones hit really hard from the start. These can be more challenging to break. So depending on how the hormones hit them they may break themselves at any time.
My test if a hen is broody enough to be trusted with hatching eggs is that she spends two consecutive nights on the nest instead of in her normal sleeping spot. If she doesn't do that she has not totally flipped. If yours has been doing that for five days consider her committed for the duration.
Some people claim success in breaking a broody using other methods. I've never tried these so I don't know how successful are. In my opinion just tossing her off the nest won't be that successful if she has really flipped over to full broody mode. The ice in the nest might work and might be easier for you to try before you commit to a cage. Here's a good link.
https://www.backyardchickens.com/threads/how-to-break-a-broody-hen.181289/#post-2176186
