My aunt passed away yesterday she had a friend/ health caregiver helped her with basic living assistance. How does my family get her belongings after police said she can continue staying there?

You sue her for the belongings: for a court order that she turn them over to you; and/or for monetary compensation for their estimated value. (Suing just for monetary compensation is procedurally easier.) In the lawsuit, she'll have a chance to prove that she has some legal entitlement to some of the belongings. Normally, she would not: being a  caregiver/friend does not itself give one any right to property--she would have to prove that either 1) there is a *written* will (one that was propertly witnessed and signed) leaving her some items after your aunt passed--oral promises that somone will inherit or receive property after death are not enforceable; or 2) that your aunt gave her the items pre-death. If she can't prove 1) or 2), she has to give them to you.
As for her staying in the unit: if you  are not the landlord, don't worry about it--just don't pay the rent for her, but otherwise, let the landlord and her work matters out. Focus on the belongings. You probably wish to bring the lawsuit for them on an "emergent" (think: "urgent" or "emergency") basis, so as to get into court faster, before she disappears with, disposses of, sells, hides, etc. some of the belongings. Ideally, hire an attorney to help you you, since adding the "emergent" factor makes the case more complicated; or alternately, if you want to do this yourself or feel a lawyer is not cost-effective, you should be able to get instructions and possibly forms from the county court clerk's office.

You sue her for the belongings: for a court order that she turn them over to you; and/or for monetary compensation for their estimated value. (Suing just for monetary compensation is procedurally easier.) In the lawsuit, she'll have a chance to prove that she has some legal entitlement to some of the belongings. Normally, she would not: being a  caregiver/friend does not itself give one any right to property--she would have to prove that either 1) there is a *written* will (one that was propertly witnessed and signed) leaving her some items after your aunt passed--oral promises that somone will inherit or receive property after death are not enforceable; or 2) that your aunt gave her the items pre-death. If she can't prove 1) or 2), she has to give them to you.
As for her staying in the unit: if you  are not the landlord, don't worry about it--just don't pay the rent for her, but otherwise, let the landlord and her work matters out. Focus on the belongings. You probably wish to bring the lawsuit for them on an "emergent" (think: "urgent" or "emergency") basis, so as to get into court faster, before she disappears with, disposses of, sells, hides, etc. some of the belongings. Ideally, hire an attorney to help you you, since adding the "emergent" factor makes the case more complicated; or alternately, if you want to do this yourself or feel a lawyer is not cost-effective, you should be able to get instructions and possibly forms from the county court clerk's office.