We went to Skiathos for a week when ds was nearly one and even without childcare had a relaxing time.
Try getting/taking a back pack - you never know, she might actually fall asleep in that with the rocking motion of walking. And even if she doesn't, at least you're getting out!
We took one of the ones with a frame (I can't remember which make as we bought it 2nd hand but like the Kelty Kids, GS90 or Karrimor Papoose on the Best Buys here), which worked well as if ds (nearly 1) went to sleep we could take him off dh's back and leave him "propped up".
Take the buggy with you when you go to markets etc - at least you'll have it for when she gets tired, and until then, it can be used to carry provisions from the market!
Get a pop up sun tent and then she can play in it with some books and stacking cups. With any luck she'll have some naps in there as well, which might give you some time to relax!
We're thinking of taking a villa later this year - we actually missed having the time with ds last year, when we were on a dingy sailing holiday and ds (then 2) was in the kiddies' club.
I'm sure you'll enjoy yourselves!