Hi Brian. Well, that's a question you've posed! As I don't know the answer off the top of my head, I had a good look round the Internet and the best I can come up with is the link below. It is regarding fishing rod hire and you have to contact them for more details. I'll leave it up to you if you want to follow it up.
Link: http://www.letitclean.co.uk/ (then click on "Fishing rod hire")
I've also discovered that many apartment complexes and hotels hire out fishing rods, so perhaps, where you're staying, they do this.
If any of you out there knows of any fishing tackle hire shops in the Adeje area, perhaps you could let Brian know.