Difference between 着 正在　and 把
我把作业写完了。 This is correct. I've finished my homework.  

我正在写得完作业。→ 我正在写作业。I'm doing my homework. 

我写着作业到完了。→ 我一直把作业写到完为止。I've been doing my homework until it was done. 

着 isn't used as much as people think, contrary to what grammar texts might explain. It's used more in written Chinese than in spoken Chinese, but that's just my opinion. Other natives might disagree. 

Hope that helps. :)