I'd probably go for either of the 2 main tourist areas. Coralejo or Caleta de Fuste.

Have you been to Fuertevenura before, if so which area do you prefer?

I've been to Corralejo before and quite like it but never to Caleta de Furste.
Took a look at El Cortillo quite a few years back and it looked like it could be a nice place once it's developed but not when we were there!

Hi ,just happened to be watching channel 4 place in the sun last night , and it was fuerteventura , I carried on watching as my sister likes it there , and I wondered what prices are like now , on the link below it was episode 30 I watched and I'm sure you could get it on 4 OD ( on demand ) like I player catch up , it was quite interesting and good to see what makes people choose differing properties and resorts , and with exchange rate being quite strong at the moment , not a bad idea to be considering a purchase .
Wouldn't mind doing the same one day , still too many days yet til retirement due !! but my ideal will be just across the water in Playa Blanca

http://www.channel4.com/programmes/a-place-in-the-sun-2015/episode-guide

Ch4 prog would be useful for you. Ferte is my favourite island have considered buying there in the past It would have been the South of the Island either Costa Calma or Jandia but there isn't a big Brit ex pat population down there if that's what you want. .. I would now opt for a rental but time is short and the World is a big place.