[verb] + 着 + (了)
[verb] + 得 / 不 + 着
着 after a verb is a result complement – a word after the verb that says how it turned out. The result here is success: the action got what it was aiming at. It is spoken, and most common in the north.
The common verbs are 找, 睡, 买, 猜, 见, 借 and 碰. With most of them 到 does the same job – 找到, 买到 – and is more neutral. 睡着 has no such partner: it is the only way to say "fall asleep".
没 + verb + 着 says it didn't succeed. 不 between verb and 着 says it can't: the potential form.
The same character as aspect 着 zhe, which marks an ongoing state (门开着). After 睡, 找, 买 and the rest, and in the 得 / 不 forms, it is zháo.
睡觉 is a verb with a built-in object (觉). The complement attaches to 睡, and 觉 drops out.