I think you're wrong in your assumption that a long-term houseguest becomes a tenant or lodger, even though the houseguest has never paid a penny in rent. But you can clarify that with a California lawyer if you choose.

If you were the owner of the dwelling you would have recourse under the lodger statute:
Quote Quoting California Civil Code Section 1946.5.
(a) The hiring of a room by a lodger on a periodic basis within a dwelling unit occupied by the owner may be terminated by either party giving written notice to the other of his or her intention to terminate the hiring, at least as long before the expiration of the term of the hiring as specified in Section 1946. The notice shall be given in a manner prescribed in Section 1162 of the Code of Civil Procedure or by certified or registered mail, restricted delivery, to the other party, with a return receipt requested.

(b) Upon expiration of the notice period provided in the notice of termination given pursuant to subdivision (a), any right of the lodger to remain in the dwelling unit or any part thereof is terminated by operation of law. The lodger's removal from the premises may thereafter be effected pursuant to the provisions of Section 602.3 of the Penal Code or other applicable provisions of law.

(c) As used in this section, "lodger" means a person contracting with the owner of a dwelling unit for a room or room and board within the dwelling unit personally occupied by the owner, where the owner retains a right of access to all areas of the dwelling unit occupied by the lodger and has overall control of the dwelling unit.

(d) This section applies only to owner-occupied dwellings where a single lodger resides. Nothing in this section shall be construed to determine or affect in any way the rights of persons residing as lodgers in an owner-occupied dwelling where more than one lodger resides.
But you seem to be a renter yourself, so that language does not seem to apply. So if you want to evict, you would presumably have to give the proper notice, then go to court to get an eviction order.

Given the cost and burden of that process, I think you should check out my initial gut instinct - that this person is a mere guest who can be tossed out any time you choose - and if a local lawyer agrees with that assessment have the landlord change the locks the next time she is out (or is institutionalized).