If you pursue the case in civil court, you will have the right to subpoena evidence from anyone who has it, including the hospital or sheriff. This is a common issue. Generally speaking, the proper procedure is to send the holder of the evidence a "Notice of Imminent Litigation and Demand for Preservation of Evidence", which would put them on notice you need that evidence for a court case. Then, once the lawsuit is filed, use a subpoena to not only get the rape kit, but also the testimony of all the people involved in collection and testing.
It may be correct that you cannot just get it by request. But I am 100% sure that a litigant in a civil suit (including the defendant) can get that evidence by subpoena.
There is a process that would allow the hospital to challenge the subpoena, or to ask for a "sealing of the evidence". However, if you are suing in civil court for rape (sexual assault) and there is a rape kit at the hospital, all of the litigants have a right to it.
An attorney you hire to conduct the lawsuit will know how to issue that subpoena.
Can you do it alone without an attorney during a lawsuit you bring yourself? Possible, but complicated.
Follow up questions?