Long term rentals are in really short supply here, and sadly there aren’t any agents working in that space.
There are a few Facebook groups like this one: https://www.facebook.com/groups/154519098019099/ but also search FB for “Long term rentals lanzarote” and you will find others.
If you’re prepared to look outside of the resort areas, you’ll find rural properties are good value for money and they often have longer periods of availability, so you can do a “deal ” for a long stay.