If you don't have room for a caregiver, why hire a caregiver? You expect her to sleep in the closet?
Either get a bigger apartment or look for other solutions. Kids 9 and 10 will be in school, there are a couple of hours a day where they will be home after school until you are home from work. In my homeland, you would give them keys to the apartment so they could get in, tell them they can have a yogurt or a slice of bread if they are hungry and please not to set the house on fire. If there is some reason you can not trust your kids in the apartment, first check if their school has an after school program, failing that, put up a note in your building or in your local grocery store for a hired grandma they should go and stay with until you are home or talk to the parents of their classmates, one of who might be a stay home. During the summer vacation you can look for other solutions such as day camp.