I argue that it's correct to say "没看我忙着呢吗？". I'm not sure if it's an regional saying or not, it's often said in the north of China though. the character 呢 here has no actual meaning, it just makes the sentence present progressive, which means "I'm being busy now." To make a sentence present progressive, you can simply put 呢 after the object of the verb, some examples listed below; 我吃饭呢。 I'm having a meal. 我看电视呢。 I'm watching TV. 我看书呢。 I'm reading a book. Hope this helps.

usually people from north would say "呢“, kind of pet phrase. an emphasis tone adverb.

呢 is generally used to soften a question or ask a question that is obvious or redundant. Likely here is was used to confront the other person in a softer less harsh sounding way.

the given sentence is incorrect

I think the '呢‘ is completely useless. Just drop it. Most mandarin speakers would say 没看我忙着吗！