Congrats on the new chick!
A hen will decide for herself... some are perpetually broody and will raise their chicks till 6 wks or so, lay eggs for a few weeks and then go back to the 'broody mood' .... others are more seasonal and go broody once a year or so. No rhyme or reason.
To encourage broody moods you can try placing fake eggs (aka golf balls! LOL) in a nest to prod their urge to sit. It doesn't always work, but sometimes. Often if one hen in a group goes broody then others may follow, folks like to joke that it is contagious, but it doesn't always happen that way.
Mark the eggs you want her to hatch and check regularly to remove eggs that others have added to her clutch. Once she starts to sit you really don't want to keep adding eggs over multiple days because they won't all hatch at the same time and it may cause her to want to stay on the unhatched eggs and neglect her already hatched chicks, or she may get off the nest to tend the already hatched chicks and the eggs left may die. Neither is a good result. So try to keep the egg due date within a day or two...
A hen will sit for well over the 21 days... they want chicks and they are determined. If eggs don't hatch they will often continue to sit till they can hatch out a new group of eggs or you give them day old chicks from the feed mill to adopt. There are ways you can discourage a broody mood also... just look up 'breaking a broody hen' in the search above to get info for that.
You may encourage them by placing a group of 'fake eggs' in a couple of the non favored boxes. I use white golf balls and place 4 to 6 in a box I want to encourage use in. It can sometimes trigger a broody urge also. Using the fake eggs will allow you to collect your real ones daily.