sleep和sleeping,分别是动词的原型和ing形式
I'm going to sleep.我去睡觉了
She's sleeping.她睡了.
asleep是形容词,表示睡着了：She's asleep.她睡着了
sleepy和sleepily,分别是形容词以及其副词形式,表示困了、累了,意思是一样的,副词用来修饰动词而已
I'm sleepy.我累了
She sleeps.她在睡.
She's sleeping.她睡了.（这两句差别不大,但是ing形式更强调这个状态.）
She's asleep.她睡着了.
She's sleepy.她困了.
但是不能 She's sleepily.
She goes to work sleepily.她满怀困意的去上班了.