I've spent some time teaching people web development (Ruby, Rails, Javascript) both as a professional trainer and as a volunteer through RailsBridge. I've seen a few motivated individuals go from non-coder to software developer in less than 1 year. Certainly you can get paid jobs writing code sooner, but these folks were doing real development in 6-9 months.

With an engineering background you already have a good foundation in problem-solving. I would recommend learning one language really well first. I like Ruby or Python for beginners. Javascript is very practical, but I agree with the other folks in this thread who compared it to C. It's lack of structure make it a hard first language. Then find a small open source project to get involved with. A lot of open source coders will happily mentor you if you put in the effort to contribute -- you can start by contributing docs and good bug reports. Get started, see if you like it. It's not easy, but it can be done. For me, it has always been fun.

By the way, I'm a she :)