You did not tell us where your Mom resided at the time of her death. As she has a house in Chicago, I presume she resided in Illinois. If that is true, you need an Illinois probate attorney.
I assume your Mother was not married at the time of her death. If there is no surviving spouse, all the children can agree as to nominating one as executor and that would be preferable to co-executors. Once the executor is appointed by the Illinois probate Judge, he or she can proceed to sell the property, administer the estate, pay your mom's debts, and divide the remaining proceeds among the beneficiaries.
It is necessary to know where your Mom was domiciled at thte time of her death. I am assuming that she was residing in Florida and had a home in Illinois. If that is the case, you would open a probate in Florida. Since there was no Will, the intestacy laws will be used to determine who receives what from her estate. As a child of the estate, one of the siblings can petition to be appointed Personal Representative. You will have to open what is called an ancillary estate in the state of Illinois. But someone must be appointed as Personal Representative first. Again, this is all if your Mom was domiciled in Florida at her death. Good Luck.