It does not matter when you will both select to get to sleep but as long as it's known that you really need to do that at some point, followed by things are clearer.
One of the popular hotels is Hotel Pirat; it is only 80 km from Fathiye or 130 km from Dalaman airlines. http://zaxx.Co.jp/cgi-bin/aska.cgi/m2tech/index.htm