1) A lease is generally terminated by the resident's death, so there should be no need to pay for rent going forward from her passing. So the deposit should not be applied to that.2) The security deposit could be used to pay any unpaid back rent (if any) or for any required repairs or exceptional cleaning required; those are the purposes of a security deposit. So if there is any money owed for these things, the landlord can take it out of the deposit. The landlord would need to prove or substantiate any amounts due.3) The deposit would be property of her estate, not a family member, and it is the estate which should try to get it back. The administrator or exectutor of the estate would be the one who contacts the landlord and attempts to recover the security deposit.