Uyurum, yes. Both uyurum and uyuyorum are acceptable.
Turkish simple present tense is not the same as the English simple present. It can be used for habitual actions, but it's not obligatory. That tense has a wide range of other important uses. You may want to take a look at: Aorist Usage.
It means "Bütün gece uyuyorum." and that implies only one night. In other words you spend your time sleeping at a whole night. But i see what you mean. The sentences "Her çocuk güzeldir." and "Bütün çocuklar güzeldir." have the same meaning but "Her gece uyuyorum" and "Bütün gece uyuyorum" do not. You might say "Bütün geceler uyuyorum." but it sounds very weird.