You can only really kite Alex Headland itself in a really strong SE when the beach is empty and no surfers in the water however,
Maroochy Rivermouth is a short walk away which is one of the most popular spots on the coast and there is also about 50km of open beaches to kite on where most of us here do downwinders.

Keep well clear of flagged areas and there are some restrictions in Caloundra (which I can't find the link to right now). The following stretches of beach are the best for downwinders.
Coolum to Maroochy river
Kawana to Currimundi (some restrictions at Currimundi)
Coolum to Sunshine beach
Stumers Creek

Noosa Rivermouth also good in NE wind (there are some restrictions there which are posted up at the site)

I suppose I should add, we don't have any transport, so any places we go will have to be foot falcon or bus

Is it ok to kite right infront of our hotel? Or is this the beach someone mentioned earlier that only works on a strong SE? Otherwise we'll just walk to maroochydore I guess.

We're going early December, if anyone sees a black/yellow/grey torch with the rider eating it, come say hello.

Another thing you could do in a Southerly is do a down winder from Maroochydore to Mudjimba or Coolum (Stumers Creek) and then get a bus back to Alex. We have taken buses before when doing a solo run.